I. Introduction

The microscopic world of atoms is governed by different rules than the classical world physicists were used to in the early twentieth century. That was the origin of the observations of scientists such as Niels Bohr and Max Planck that later led to the principles of Quantum Mechanics. However, the discoveries of quantum mechanics remained a theory for years, until it led to the inevitable question: how can we create real-life applications using the most counterintuitive properties of quantum mechanics? A class of applications has been featured that employ quantum mechanical systems as sensors for numerous physical quantities ranging from time and frequency, to magnetic and electric fields, to temperature and pressure. The mechanism of work of quantum sensors capitalizes on the central weakness of quantum systems, their strong sensitivity to external disturbances, in addition to the one consistent yet odd fact about the congruence of atoms [1]. To gain sight of how one form of quantum sensors functions, we will consider clocks. Let’s say we need to build a clock; what criteria should be satisfied by a good clock? To answer this question, we could think about the basic pendulum clock. In 1583, Galileo Galilei discovered that a pendulum of the same length always takes the same time swinging back and forth i.e. completing one oscillation. If the periodic time -the time it takes the pendulum to complete one full oscillation- was known to be exactly one second, counting 60 oscillations would make 1 minute, and so on. Therefore, a good oscillator is a crucial point in building an accurate clock. Nonetheless, this gives room for some complications in relying on the pendulum as a precise clock. If, for example, a clockmaker in Egypt made a pendulum clock of the same length as one made in China, can it be certain that both robes are identical on the molecular level? And if so, can the stretch and variations due to external factors be prevented to keep track of time? Some issues arise with the debate about “the perfect oscillator”. Later on, LC Oscillators played a distinctive role in the history of clock-making. Several wristwatches as well as radios and televisions depended on the oscillations of an LC circuit. However, the poor frequency stability of the phase shift oscillator, the difficulty to control the amplitude of oscillations, and the lost energy of the system due to resistance all prevent LC oscillators from being reliable time trackers. Later in history, Pendulum and LC clocks were then succeeded by quartz crystal oscillators. They were based on the phenomenon of piezoelectricity discovered by the French physicist Pierre Curie. It resonated at a nearly constant frequency when an electric current was applied. The invention of the quartz crystal clocks was an improvement in time measurement for humanity. However, Quartz Oscillators’ resonance frequency depends on the size and shape of the crystal, forbidding them from being ideal frequency standards since no two crystals can be exactly alike. Atoms, on the other hand, have resonances that are stable over time and space; a hydrogen atom in Egypt resonates at the same rate as one in the arctic. Also, if you were given one atom today and another that is 1000 years old, there is no way to differentiate between the two atoms. This happens because each chemical element and compound absorbs and emits electromagnetic radiation at its particular characteristic frequencies i.e. energy levels. That explains the fact that as atomic transitions are identical in the same type of atom, a group of atomic oscillators resonates at precisely the same rate, unlike a group of quartz oscillators. Atoms also do not wear out with time, unlike electrical and mechanical resonators. Atoms do not change their properties over time, contrasting with crystals that are affected by environmental factors. [2] These limitations of various oscillators along with the quest for better accuracy led to the development of the atomic oscillators, and therefore, atomic clocks. At first sight, a minor shift in a clock, one that is 1 millionth of a second a day, for example, may not seem catastrophic in human daily activities. But the accuracy of satellite navigation systems and Global Positioning Systems (GPS) highly depend on the proper synchronization of time. A difference in 1 microsecond may cause a positioning error of approximately 300 m. If that change occurred regularly, a critical part of how our world functions today would soon be useless. This happens due to the effect of time dilation on how satellites experience time while moving with a speed of 14,000 km/h around the earth in the lower gravity in space, the two factors that greatly affect time dilation according to Einstein’s theory of special relativity. [3]

II. Early History of Quantum Mechanics:

As the advancements in physics continued over time, it did not take long for physicists to realize how odd atoms are. Classical Physics described the world fairly well. The calculations and predictions were mostly accurate when dealing with a classical problem. For example, specifying the initial position and the velocity of a ball would be enough to calculate the time it took to reach its final position on a plane. Classical Physics could successfully be applied to many systems, but would it succeed to describe the tiniest system of them all-the atom? It turned out that classical physics must fail for the atom. If we compare any two classical systems together, there will always be a certain percentage of the difference between them. This fact contrasts with the hydrogen atoms which are, as discussed in section (1), identical over time and place. From there, the idea that there must be another way to describe the behavior of atoms came.

i. The Bohr’s Model of the Atom:

ii. The Origin of Spectra:

This emission could not be explained using classical mechanics. It was not believed that an electron orbiting in its usual orbit can emit light since it implied a gradual loss of energy that would gradually make that orbit smaller. Eventually, it would collapse into the nucleus, which we know does not happen. Yet, light is still emitted. To explain this, Bohr put three postulates to sum up his model:

1. The electron revolves in stable orbits around the nucleus without radiating any emissions, counter to what classical electromagnetism suggests. The stable orbits are called stationary orbits and are at discrete distances from the nucleus. The electron cannot exist in any space nor vacant orbits in between these discrete ones.
2. The stationary orbits are at distances at which the angular momentum of the electron is an integer multiple of the reduced Plank’s Constant: $m_e\nu r = n \hbar$, where $n \in \mathbb{Z}$, $n$ is called the principal quantum number, and $\hbar = h/2\pi$. The lowest allowed number of $n$ is 1. It is the closest the electron can get to the nucleus. Bohr could calculate the energies of the allowed orbits of the hydrogen atom and other simple hydrogen-like atoms and ions. In these orbits, the electron can revolve without energy loss or radiation. [5]
3. Electrons can gain or lose energy by jumping from one of the allowed orbits to another while absorbing or emitting radiation with a frequency v that is determined based on the energy difference of the levels according to the relation: $\Delta E = E_2 - E_1 = h\nu $, where $h$ is Plank’s constant.

Bohr assumed that during a quantum transition a discrete amount of energy is radiated, just as in Einstein’s theory of the Photoelectric Effect. However, Bohr stuck to Maxwell’s theory of the electromagnetic field contrary to Einstein. [6] Bohr’s condition that the angular momentum must be a multiple of $\hbar$ was later reinterpreted by de Broglie; the electron is described by a wave and a whole number of wavelengths should be along the circumference of the orbit.

$$n\lambda = 2 \pi r$$

(2.1)

According to de Broglie, matter particles behave as waves. Thus, the wavelength of the electron is:

$$\lambda = \frac{h}{m \nu}$$

(2.2)

This implies that:

$$\frac{nh}{2 \pi} = m \nu r$$

(2.3)

Where $m \nu r$ is the angular momentum of the spinning electron and $\frac{h}{2 \pi}$ is $\hbar$. Writing $l$ for the angular momentum, the equation becomes:

$$l = n \hbar$$

(2.4)

Which is Bohr’s second postulate. And since n is an integer, we can conclude that angular momentum is quantized.

But how well does this model hold in explaining the atomic spectrum? Bohr tried to answer that question using the hydrogen spectral lines like those in figure (2). In classical mechanics, a set of different initial positions may result in the same final answer. For example, balls falling with different initial velocities end up with a final velocity of zero. Each one started with certain energy but ended up with the same final energy. The energy dissipated. To describe a classical attractor, energy must be dissipated, but in the macroscopic world, energy is conserved. So, electrons cannot dissipate their energies, but they need to get to the same final state. Bohr thought of differential equations in order to solve this problem. To reach the same final state in a differential equation, Bohr tried to restrict the initial states. As mentioned in section (2.1), the force binding the atom together is the electrostatic force. According to Coulomb’s law, the magnitude of that force can be described by the famous equation: $F = \frac{KQ}{r^2}$, where $K$ is Coulomb constant, $Q$ is the total electric charge, and $r$ is the distance. Since $K = \frac{1}{4 \pi \varepsilon_0}$, where $\varepsilon_0$ is the electric constant, or permittivity of free space, the equation can be rewritten as:

$$F = \frac{Q}{4 \pi \varepsilon_0 r^2}$$

(2.5)

And since

$$F = \frac{Q}{4 \pi \varepsilon_0 r^2} = \frac{m_e \nu^2}{r}$$

(2.6)

We already set $l = m_e \nu r = n \hbar$, so $\nu$ can be written as: $\nu = \frac{n \hbar}{m_e r}$.

Therefore,

$$r = \frac{n^2 \hbar^2}{m_e Q^2} \, 4 \pi \varepsilon_0$$

(2.7)

Thus, Bohr could restrict the initial conditions of the electron. Now, to calculate the energy of the system:

$$E = \frac{m_e \nu^2}{2} - \frac{Q^2}{4\pi\varepsilon_0r}$$ $$= \frac{m_e^2 \nu^2 r^2}{2 m_e r^2} - \frac{Q^2}{4 \pi \varepsilon_0 r}$$ $$= - \frac{m_e Q^4}{n^2 32 \pi^2 \varepsilon_0^2}$$

(2.8)

The energy is negative due to the fact that it is a binding force. It is also quantized meaning that only certain energies are allowed and therefore, only certain transitions in the atom.

iii. The Electron doesn’t have a Fixed Position:

Restricting the initial positions was enough to produce spectra. However, these restricted positions may not stand true in each atomic phenomenon. For example, chemical reactions yield the same compounds each time. Two molecules of hydrogen with one oxygen molecule would always react to produce a water compound. Atoms with quantized initial conditions have fixed energies, but varying phases. And although the phase varies, atoms still scatter in the same manner. Scattering happens when particle-particle collisions occur between atoms, molecules, chemical compounds, and photons. It is the main factor chemical reactions such as the one discussed above happen. When the three atoms collide, they form the hydrogen bonds necessary to get a water molecule. A possibility might be that electrons move so fast that their phases average out. But the fact is that chemical reactions do not depend on the phase of the electron; chemical reactions in solids and liquids produce the same yields as gases. Reaction rates in solids and liquids are large because of their high density, and rates are comparable to electron energy in chemical reactions. If the phase has any effect, it should show up. But in fact, no such effect is seen. We cannot think of electrons as having restricted initial positions. They behave as if they do not actually have a classical definite orbit around the nucleus where they have positions and phases. The remaining option would be that electrons do not have well-defined positions in the atom. That means that the classical pictures of the atom as a dense nucleus surrounded by orbitals on which electrons are present are wrong, and cannot explain the majority of how atoms behave in nature.

III. The Quantum Principles and Mathematical Background

i. The Schrödinger Equation

When thinking about the electron, we would find it to be an analogy to the above problem. However, quantum mechanics approach it quite differently to cope with the electron’s nature. In this case, we want to solve the electron’s wavefunction $\Psi(t,x)$, and to solve it, we need the Schrödinger Equation:

$$i \frac{\delta \Psi}{\delta t} = -\frac{1}{2m} \frac{\delta^2 \Psi}{\delta x^2} + V(x) \Psi$$

(3.1)

Where $V(x)$ is the potential the particle experiences.

This wave function is a complex-valued function that is interpreted as a probability density i.e. the probability of finding the particle between two points a and b at time t given by the formula: $\int_{a}^{b} \Psi^* (t,x) \Psi (t,x) dx$.

To get the wave function $\Psi (t,x)$, we need to find a way to solve equation (3.1). Since we have two variables, the equation should be solved using a partial derivative, but to avoid it, we can assume that there is a function $\tilde{\Psi}$ , such that:

$$\left ( -\frac{1}{2m} \frac{\delta^2 \tilde{\Psi}} {\delta x^2} + V(x) \tilde{\Psi} \right ) = E_0 \tilde{\Psi}(x)$$

(3.2)

Note that $\tilde{\Psi} (x)$ is only a function of $x$ and not $t$. We can find a solution to the Schrödinger equation of the form $\Psi (t,x) = F(t) \tilde{\Psi} (x)$ if we could describe $F(t)$ in terms of $E_0$.

Since we need the equation to be in the form of $\Psi (t,x) = F(t) \tilde{\Psi} (x)$, we can write a new equation using the derivative multiplication rule:

$$i \frac{\delta}{\delta t} (F(t) \tilde{\Psi} (x)) = iF(t) \frac{\delta \tilde{\Psi}}{\delta t} + i \frac{\delta F}{\delta t} \tilde{\Psi} (x)$$

(3.3)

Since $\tilde{\Psi}$ is only a function of $x$, $\delta \tilde{\Psi} = 0$, thus:

$$i \frac{\delta}{\delta t} (F(t) \tilde{\Psi}(x)) = i \frac{\delta F}{\delta t} \tilde{\Psi} (x)$$

(3.4)

$$\therefore - \frac{1}{2m} \frac{\delta^2}{\delta t^2}(F(t) \tilde{\Psi}(x)) = -\frac{F(t)}{2m} \frac{\delta^2 \tilde{\Psi}}{\delta x^2}$$

(3.5)

$$-\frac{1}{2m} \frac{\delta^2}{\delta t^2} (F(t) \tilde{\Psi}(x)) + V(x) (F(t) \tilde{\Psi} (x))$$ $$= F(t) \left ( \frac{-1}{2m} \frac{\delta^2 \tilde{\Psi}}{\delta x^2} + V(x) \tilde{\Psi} (x) \right )$$ $$= F(t) \cdot E_0 \tilde{\Psi} (x)$$

(3.6)

Therefore,

$$i \frac{\delta F}{\delta t} = E_0 F (t)$$

(3.7)

This is now a normal differential equation, by solving it we would get that:

$$F(t) = e^{-iE_0t}$$

(3.8)

The separation of variables has turned a partial differential equation into a differential equation (3.7). Equation (3.2) is the time-independent Schrödinger equation. It cannot be solved further without a specified potential $V(x)$. For a variety of potentials, the solution can take three forms:

• The first type of solution is the stationary state. Although the function $\Psi(t,x) = \psi (x) e^{-iEt}$ depends on time, the probability density

  $$\left | \Psi(t,x) \right |^2 = \Psi^*\Psi = \psi^* e^{+iEt} \cdot \psi e^{-iEt} = \left | \psi(x) \right |^2$$

  (3.9)

  does not; the time dependence cancels out. Therefore, every expectation value from this probability density is constant in time. We may use $\psi (x)$  instead of $\Psi(t,x)$, however, we cannot refer to $\psi (x)$ as the wave function since the correct wave function should always carry the time-dependent factor.

• They are states of definite total energy. The total energy is called the Hamiltonian in classical mechanics, it is written as:

  $$H(x,p) = \frac{p^2}{2m} + V(x)$$

  (3.10)

  This expression has a corresponding Hamiltonian Operator. An operator in linear algebra is used to transform a function into another function, just like how a function transforms a variable into another. When we use a substitution for $p$ as: $p = -i \frac{\delta}{\delta x}$ (the position operator), the Hamiltonian Operator is, therefore:

  $$\hat{H} = - \frac{1}{2m} \frac{\delta^2}{\delta x^2} + V(x)$$

  (3.11)

  Thus, the time-independent Schrodinger Equation can be rewritten as:

  $$\hat{H} \psi = E \psi$$

  (3.12)

  And the expectation value of the total energy is

  $$\left \langle H \right \rangle = \int \psi^* \hat{H} \psi dx = E \int \left | \psi \right |^2 dx = E \int \left | \Psi \right |^2 dx = E$$

  (3.13)

  Moreover,

  $$\hat{H}^2 \psi = \hat{H} (\hat{H} \psi) = \hat{H} (E \psi) = E (\hat{H} \psi) = E^2 \psi$$

  (3.14)

  hence:

  $$\left \langle H^2 \right \rangle = \int \psi^* \hat{H}^2 \psi dx = E^2 \int \left | \psi \right |^2 dx = E^2$$

  (3.15)

  The variance of H is

  $$\sigma_H^2 = \left \langle H^2 \right \rangle - \left \langle H \right \rangle^2 = E^2 - E^2 = 0$$

  (3.16)

  If the variance is zero, then the distribution has zero spread, meaning that every member of the sample shares the same value. In conclusion, the separable solution has the important property every measurement of the total energy is certain to return the eigenvalue E.

• The general solution is a linear combination of separable solutions. The time-dependent Schrodinger equation has an infinite set of solutions, which we collectively refer to as $\left \{ \psi_n(x) \right \}$, each with its energy separation constant $\left \{ E_n \right \}$; thus, there is a different wave function for each allowed energy:

  $$\Psi_1 (x,t) = \psi_1 (x) e^{-iE_1t} \, , \, \Psi_2 (x,t) = \psi_1 (x) e^{-iE_2t} .......$$

The time-dependent Schrödinger Equation has the property that any linear combination of these solutions is also a solution. We can now construct a more generalized solution of the form:

$$\Psi (x,t) = \sum_{n=1}^{\infty} c_n \psi _n (x) e^{-iE_nt}$$

(3.17)

Every solution to the time-dependent equation can be written in this form. But what do the coefficients $\left \{ c_n \right \}$ physically represent? $\left | c_n \right |^2$ is the probability that a measurement of the energy would yield one of the allowed energies $E_n$ values. Since it is a probability, the sum of these probabilities is one.

As a result of equation (3.17), the probability of finding a particle is now time-dependent whenever we have a superposition. Thus, this variation in the probability density over time is what gives rise to an oscillator, which could be extended to various other applications such as clocks in our case.

$$\sum_{n=1}^{\infty} \left | c_n \right |^2 = 1$$

(3.18)

And the expectation values are:

$$\left \langle H \right \rangle = \sum_{n=1}^{\infty} \left | c_n \right |^2 E_n$$

(3.19)

Since the constants $c_n$ are time-independent, so is the probability of getting certain energy and the expectation value of H.

Now, we can see how the probability density is going up and down which can provide us with an oscillator. We can also make good use of the fact that these eigen energies are very constant in quantum mechanics, and therefore, may be useful in building a system with no error.

ii. The Particle in a Box Problem

Particles tend to seek low potential, so the probability density (i.e. the probability of finding the particle outside the box) is zero. However, inside that box, where $V(x)=0$, the time-independent Schrodinger equation becomes:

$$- \frac{1}{2m} \frac{\delta^2 \psi}{\delta x^2} = E \psi$$

(3.20)

Or

$$\frac{\delta^2 \psi}{\delta x^2} = -k^2 \psi$$

(3.21)

Where $k = \sqrt{mE}$. Equation (3.21) is the equation for the famous simple harmonic oscillator; its general solution is:

$$\psi (x) = A \sin kx + B \sin kx$$

(3.22)

Where A and B are arbitrary constants. The value of each of these constants depends on the boundaries of the problem which typically implies that both $\psi$ and $\frac{\delta \psi}{\delta x}$ are continuous, but since the potential is infinity outside of the box, only the first condition applies.

The continuity of the function $\psi (x)$ requires that:

$$\psi(0) = \psi(a) = 0$$

Solving for A and B in equation (3.22),

$\psi (x) = A \sin kx$

(3.24)

As a result, $\psi (a) = A \sin ka$ sinaAka= . This leaves us with two possibilities, either $A=0$ (then we would have a non-normalizable solution), or $\sin ka = 0$ which means that

$$ka = 0, \pm \pi, \pm 2 \pi, .....$$

We realize that $k$ cannot be equal to zero (because of the function normalization). So, finally, we can find that

$k_n = \frac{n\pi}{a}$ , where $n \in \mathbb{Z}$

Thus, the possible values of E:

$$E_n = \frac{n^2 \pi^2}{2ma^2}$$

(3.25)

If we were talking in a classical context such as the cart example proposed at the beginning of the section, any energy would have been allowed. In the quantum realm, however, the particle must acquire one of the allowed eigenvalues. To find the value of $A$, we need to normalize the function $\psi$

$$\int_{0}^{a} \left | A \right |^2 \sin^2 (kx) dx = \left | A \right |^2 \frac{a}{2} = 1$$

(3.26)

As a result, $\left | A \right |^2 = \frac{2}{a}$, and since the phase of $A$ does not have any significance, we can describe it as $A = \sqrt{2/a}$. Inside the box, the solutions are:

$$\psi_n (x) = \sqrt{\frac{2}{a}} \sin \left ( \frac{n \pi}{a} x \right )$$

(3.27)

Again, the time-independent Schrodinger equation gave an infinite number of solutions. To visualize them, figure (5) gives an illustration of the first three solutions. The wave looks like a rope of length a been continuously shaken so that it gives these sinusoidal graphs.

The first graph representing $\psi_1 (x)$, the lowest energy state, is called the ground state. The others with increasing energies are called excited states. Collectively, these functions have some interesting properties:

• They alternate between even and odd functions.
• They are mutually orthogonal.

$$\int \psi _m (x) * \psi _n dx = 0 \, , \, m \neq n$$

The previous section discussed the basic ideas and the math behind the Schrodinger equation. However, in the quest of making quantum sensors, a particle in a box system is not what we need. The reason behind that is that even if, in theory, we could make that box, there would still be variations in some factors such as the box’s length. Again, we cannot guarantee that it would be as identical on the molecular level as we would like it to be to make accurate measurements. Fortunately, we still can use this system to make an analogy; the particle can be the electron, and the box -in which this electron is confined- is the atom. Recalling the fact that the eigenvalues which were calculated in section (3.2) are very constant in quantum mechanics, as well as the imposed argument in section (1) that atoms are perfectly identical, we can conclude that the atom is a decent thing to utilize while building sensors, especially clocks.

iii. The Role of Probability in Accurate Measurements

What we want to do is toss these coins as many times as it takes to figure out which coin we have out of the two. If we did one toss and got tails, we would be 100% certain that it is coin 2. But if we got heads, there would be a certain possibility of making a mistake when deciding on either coin. For example, how likely is it to be wrong if we decided it was coin 1? In this case, we would make an error of 0.5. Supposing we did two tosses and got heads on each of them, the error is now $(0.5)^2 = 0.25$. Ignoring the tales, on the tenth toss, the error would have been $(0.5)^{10} = 2^{-10}$, and on the 100^th toss, it would have reached an error of $2^{-100}$. So, with more trails and more collected data, the probability of making a mistake reduces exponentially.

To make it clearer, let’s consider the following example. Given some probabilities: $$P_1 = H \, , \, P_2 = T$$

If we made $N$ tosses and wanted to calculate the expectation value:

$$S = \frac{\left ( \sum_{1=0}^{N} T_i \right )}{N}$$

(3.28)

The crucial question is: how likely is it to be on the side of coin 1 vs. coin 1? To answer this question, we shall calculate the expectation value of each coin.

$$\left \langle S_{C_1} \right \rangle = P_1 - P_2$$

(3.29)

$$\left \langle S_{C_2} \right \rangle = P_3 - P_4$$

(3.30)

The next step is calculating the variance ($\sigma$) of both distributions, then getting the final probability expression:

$$e^{\frac{-\left ( S_{exp} - \left \langle S \right \rangle \right )^2}{\sigma}}$$

(3.31)

To sum it up into simple steps, to know the probability of your measurements:

1. Get the experimental results.
2. Calculate the expected values.
3. Calculate both the mean value, using equation (3.28), and the variance of the distribution.
4. Plug the preceding information in equation (3.31) to calculate the probability.

What the answer would tell you is the probability of making a mistake if you chose coin 1 vs. coin 2 in the previous example. With more experimental data, the probability could potentially reach, for example, $e^{-18}$ which is a significantly small number for making an error. As the number of trials increases, the deviations from the probability distribution get extremely small, and therefore, we can get accurate results from something that is probabilistic such as quantum mechanics.

IV. Allowed Atom Types and The Mechanism of Atomic Clocks

In the previous sections, we discussed the necessity of the presence of an oscillator in order to build a clock. It also elaborated on how atoms are decent candidates for oscillators due to their unique nature compared to other methods such as pendulums and LC circuits. Although the efficiency of atoms in this role is undoubted, some factors must be taken into consideration when choosing the appropriate atom to build the clock. Some of these factors will be further discussed in this section, as well as some real-life examples.

i. Properties of the Atoms used in Atomic Clocks

When considering what it would take for an atom to be useful in time measurement, we would need to see what superposition theoretically means in this case.

$$\Psi (t,x) = \frac{1}{\sqrt{2}} \left ( \Psi_1 (t,x) + \Psi_2 (t,x) \right )$$

(4.1)

We need to look at the math to see how it is supposed to keep track of time. If the probability density contains $\cos \left ( \left ( E_2 - E_1 \right ) t \right )$, knowing that $\left ( E_2 - E_1 \right )$ is a constant, counting the number of oscillations the wave would take to reach from $t=0$ to $t=2 \pi$ would predict how good of a clock we have.

$$(E_2 - E_1)t = 2n \pi$$

(4.2)

To have an accurate clock, n needs to be as large as possible, meaning that there must be a large number of oscillations within the range from $0$ to $2 \pi$ to make an accurate measurement as explained by the nature of probability in section (3.3). As a result, t has to be as large as possible to guarantee this large n number to happen. Therefore, the maximum time $T_{max}$ needed to build an atomic clock is set by the life time of the atom.

ii. Types of atoms

Based on the previous ideas, hydrogen atoms cannot be used in estimating accurate time since it decays from $E_2$ to $E_1$ in a nanosecond. This explains why Hydrogen Maser clocks cost less due to their poor long-term accuracy, unlike Cesium Clocks for example which come at a significantly higher price and accuracy. Some examples of suitable atoms include:

1. Cesium:

Some Cesium properties make it a good option for an atomic clock. For example, whereas hydrogen atoms move at a speed of 1,600 m/s at room temperature, cesium moves at 130 m/s due to its significantly larger mass. [10] [11] Cesium also has a frequency of 9.19 GHz which is larger than other elements such as hydrogen with a frequency of 1.4 GHz. [11] This high frequency provides cesium with a greater ability to measure time to a very accurate level as discussed in sections (4.1) and (3.3).

2. Rubidium:

Some things that rubidium clocks are superior to cesium clocks because of are their small size and low cost. However, they do not have the long-term stability cesium clocks acquire. That is due to their relatively lower frequency of 6.8 GHz. [11]

iii. Mechanism of Work

Inside a cesium atomic clock, cesium atoms are channeled through a tube and radio waves. If this frequency is precisely 9.192.631.770 cycles per second, then cesium atoms will "resonate" and their energy state will change. A detector at the end of the tube counts the number of cesium atoms with altered energy levels that reach it. More cesium atoms reach the detector the closer the radio wave frequency is approaching 9,192,631,770 cycles per second. The detector provides the radio wave generator with information. It synchronizes the radio waves' frequency with the maximum number of cesium atoms striking it. Other electronic components within the atomic clock measure this frequency. As with a single pendulum swing, a second is deducted when the frequency count is reached. The NIST-F1 cesium atomic clock is capable of producing a frequency so exact that its daily time error is approximately 0.03 nanoseconds, which equates to a loss of one second per 100 million years.

V. Conclusion

To sum up, quantum censoring is one of the most promising applications of the unconventional properties of quantum mechanics. Research in this field was able to open up new dimensions for us in topics that were once considered fixed and non-negotiable, such as the definition of time, starting with Newton’s ideas about absolute time, to Einstein’s theory of special relativity. The quest for the perfect clock has been going on for centuries, starting with the pendulum clock which utilized Galileo’s work on the pendulum oscillations, to mechanical clocks, LC oscillators, Quartz clocks, and finally Atomic Clocks. All of these examples had one thing in common: a stable oscillator. As the work in quantum physics advanced over the years, scientists knew the fact that atoms of the same element are identical over space and time provided an ideal oscillator for a clock. This continuous research included the study of the electron’s wave-particle duality, hydrogen spectrum, the Schrodinger equation, and more advanced concepts of laser and quantum physics. Being able to measure time to the accuracy of approximately $10^{-19}$ at first seemed imaginary, especially with the uncertainty that comes along with studying quantum physics. Interestingly, the exact properties making quantum mechanics a probabilistic science in its measurements are the ones enabling it to measure with unprecedented accuracy. This preciseness has made concepts that were once theoretical, such as special relativity and quantum entanglement, a daily phenomenon we need to deal with. Furthermore, this accuracy allows us to explore different other measurements including distance, using the constant speed of light and simple equations such as $d = \nu \cdot t$. This is a simple concept on which crucial systems are based including the SONAR which is used in several medical fields and technologies. Determining precise locations using the Global Positioning Systems (GPS) also depends on measuring the distance between the satellite and the receiver. This depends on the accurate measurements of the radio signals’ travel time, which travel at the speed of light at a fraction of the second. The slightest mistake, even as small as one in millionth part of the second, could cause catastrophes to military services, astronomical research, rocket launching, airlines, and more. The use of precise time can be extended to measure the minuscule variations in the gravitational acceleration on the earth’s surface, which could lead us to learn more about the history of the geosphere and the formation of the crust. It also helps with determining the motion of plate tectonics, and therefore significantly improved our ability to predict the exact location and time of earthquakes and volcanic eruptions. Future research focuses on utilizing the concepts we learned so far about quantum mechanics and the nature of the atom in medical fields, especially in early disease detection, and high-energy production. Quantum mechanics is still somewhat ambiguous to us. Some discoveries and advances do not follow up with our comprehension, but they still, however, reveal a lot about what our universe is, what its past was, and how the future will be.

I. Introduction

Parkinson's disease is a neurological disease that causes unintended or uncontrollable movements, like shaking, stiffness, and difficulty with balance and coordination. More than 10 million people worldwide live with PD [1] . Symptoms usually begin gradually and worsen over time. As the disease progresses, the patient may have difficulty walking and talking. He may additionally have mental and behavioral changes, memory difficulties, fatigue, sleep problems, and depression. Such a disease was treated in many ways, but none of them were that effective. So, researchers started to use optogenetics to treat the disease. Scientists such as Steinbeck et al. applied optogenetics to study the function of transplanted dopamine neurons in a mouse model of Parkinson’s disease. Other scientists used optogenetics as a research tool to inspire the next generation of DBS-based therapies. In this research paper, we will go deep into how optogenetics can treat Parkinson's disease. We will focus on the application of optogenetics as a research tool to inspire the next generation of Parkinson's disease treatments. We will discuss how knowledge about cell type diversity in the basal ganglia has shaped clinical approaches to treating Parkinson's disease. Also, we will show some recommendations for future research inspired by the results.

II. The Incurable Disease

i. Brief about the body movement: the dopamine function

ii. Parkinson disease

1. Mechanism

2. Motor symptoms

1. Tremor: it is an involuntary rhythmic shaking or slight movement of the body. The tremor mainly occurs in the hands. Moreover, it can affect the chin, lips, face, or legs.
2. Slowed movement (bradykinesia): the patient impairs mobility over time, making routine tasks become complex and time-consuming. Bradykinesia comprises two terms akinesia and hypokinesia. Bradykinesia is the slowness of a performed movement, whereas akinesia is a lack of spontaneous movement [5] .
3. Rigidity: it is defined as arm or leg stiffness beyond what would result from normal aging or arthritis. The rigidity can occur on one or both sides of the body, resulting in a reduced range of motion.
4. Dystonia: it is an involuntary muscular contraction that occurs when the brain instructs the muscles to tighten and move even when you do not want them to. Dystonia, or severe muscular cramping, is a common symptom of Parkinson's disease. Dystonia is most commonly found in the feet, hands, neck, or face.

III. The Glimmer of hope

ii. Types of treatments for Parkinson’s disease

Parkinson’s Disease has been treated in many different ways over years. The illustrated treatments have shown great results; we will show the side effects for each treatment.

1. Pallidotomy & Thalamotomy

Pallidotomy is recommended for people who have severe Parkinson's disease symptoms. Pallidotomy can prevent rigidity and dyskinesias induced by some Parkinson's disease drugs [7]. Pallidotomy is performed by inserting a wire probe into the Globus pallidus, which becomes hyperactive in Parkinson's patients. Applying lesions to the globus pallidus aids in regaining the equilibrium required for regular movement. This method aids in the removal of medication-induced dyskinesias, tremors, muscular stiffness, and the progressive loss of spontaneous movement. Nevertheless, pallidotomy can cause memory loss, visual tract damage, etc. Thalamotomy is similar to Pallidotomy. Instead, it eliminates a little section of the brain known as the thalamus. However, it is only conducted on one side of the brain, which is ineffective if the patient has tremors in both hands because only one hand will recover [8].

2. Functional electrical stimulation (FES)

3. Transplantation of dopaminergic neurons

Research for a dopamine cell replacement treatment for Parkinson's disease has taken more than three decades. Previous efforts to establish the mechanism of action of transplanted dopamine neurons often entailed the use of poisons to destroy the grafted cells [12]. The problem with such approaches is that they make it impossible to differentiate between different potential cell functions, such as dopamine release, immunomodulation, or paracrine factor secretion, and to clarify cause-and-effect correlations.

4. Deep Brain Stimulation

IV. Optogenetics & electrical Deep Brain Stimulation: A Close Look

i. Historical Appearance

Alim Benabid, who found in the late 1980s that electrical stimulation of the basal ganglia reduced Parkinson's disease symptoms, is largely credited with the invention of contemporary deep brain stimulation (DBS). The later discovery of DBS has transformed the therapy of movement disorders [13]. In 2002, deep brain stimulation (DBS) for the treatment of Parkinson's disease (PD) was granted approval by the US Food and Drug Administration (FDA). Since then, approximately 40,000 patients with Parkinson's disease (PD) and essential tremor have been treated with DBS, which employs a device like a pacemaker to administer continuous electrical stimulation to specific brain regions. Despite being an approved therapy for Parkinson's, the exact mechanism for DBS is not fully understood and protocols rely on clinical outcomes for optimization of the electrical strength and polarity of the neurostimulator. Thus, as with any other treatment, scientists improved DBS stimulation using a tool called optogenetics [14].

ii. Optogenetics

Optogenetics is a precise method of accurately regulating and monitoring the biological processes of a cell, group of cells, tissues, or organs with high temporal and spatial precision [15]. According to Francis Crick, influencing one kind of brain cell while leaving others unaffected is the major issue facing neuroscience [16]. Optogenetics has made it possible to precisely regulate one neuron without impacting the others. Optogenetics works by controlling a particular neuron with light. It is based on taking special genes from algae and other single-celled organisms. These single genes are known as microbial opsins. Three major classes of rhodopsins have been developed as optogenetic molecular sensors: channelrhodopsins, halorhodopsins, and archaerhodopsins. They are added to the genetic code of the neurons to produce proteins that make them sensitive to light. The most popular opsin is called channelrhodopsin (ChR2). This opsin comes from the green algae Chlamydomonas reinhardtii [18]. It is only affected by blue light. So, neurons that have ChR2 will only be affected by blue light.

iii. the mechanism of Optogenetics deep brain stimulation

1. Network modelling

The majority of existing models of the basal ganglia are static models, representing the inputs and outputs of the component nuclei as firing rates. Within the bounds of our understanding of the topography of the connections between neurons and the cellular properties involved, we have carried out computer simulations of conductance-based models of the subthalamopallidal circuit to investigate such dynamic interactions. Conductance-based models are the simplest possible biophysical representation of an excitable cell, such as a neuron. The protein molecule ion channels are represented by conductances and its lipid bilayer by a capacitor [19]. This model consists of 10 external Globus Pallidus (GPe) and 10 subthalamic neurons (STN). Each STN neuron sends a connection to one GPe neuron, and each GPe neuron sends a connection to the STN neuron from which it gets a connection and to its two nearest neighbors. A set of ordinary differential equations based on conductance are used to describe each neuron. Membrane potential obeys (1),

$$C\frac{\mathrm{d} V}{\mathrm{d} t} = -I_L - I_K - I_{Na} - I_T - I_{Ca} - I_{AHP} - I_{syn} + I_{app}$$

(1)

Variables
$I_{app}$: a constant external applied current; $I_L$: leak current; $I_{Na}$: sodium currents are essential for the initiation and propagation of neuronal firing; $I_K$: potassium currents involved in the regulation of membrane excitability and the control of the firing pattern. (To review how to calculate the other variables review [20] )

2. Optogenetics stimulation

During the technique of optogenetics stimulation, channelrhodopsin ($ChR2$) is used as a sodium light- activated channel to stimulate neurons by depolarization. Furthermore, the halorhodopsin ($NpHR$) functions as a channel triggered by chlorine light to decrease neuronal activity by hyperpolarizing it. The idea of optogenetic stimulation comprises two types: excitation and inhibition. The ChR2 photocurrent is given by

$$I_{ChR2} = g_{ChR2} (V-V_{Na}) (O_1 - \gamma O_2)$$

(2)

Here is a brief list of the notation used: Variables
$V$: Volume (dl); $g_{ChR2}$: the maximal conductance of the $ChR2$ channel in the $O_1$ state; $V_{Na}$: the reversal potential; $\gamma$: the ratio of conductance in $O_1$ and $O_2$ states.
To know more about the equations of $O_1$ and $O_2$, review [21] The inhibition photocurrent is given by

$$I_{NpHR} = g_{NpHR} (V - V_{cl})0$$

(3)

Variables
$V_{Cl}$: the reversal potential of the chlorine channel; $g_{NpHR}$: is the maximal conductance of the $NpHR$ channel in the $O$ state.
To read more about the equation, review [21]

3. Efficacy

To compare electrical and optical stimulation, we use RMS (root mean square) current delivered to the network averaged over time T: where $I_x$ can be $I_elec$, $I_{ChR2}$ or $I_{NpHR}$

$$I_{x}^{RMS} = \frac{1}{10} \sum_{i=1}^{10} (\frac{1}{T} \int_{0}^{T} I_{x,i}^2 dt)$$

(4)

In order to compare the efficacy of optogenetic stimulations with that of electrical stimulation, we consider the minimal IRMS necessary to suppress the beta activity below the threshold. The minima are computed for a variety of light pulse durations and stimulation intensities. These values were normalized to:

$$Efficacy_{NpHR} = \frac{(I_e^{RMS} - I_{NpHR}^{RMS})}{I_e^{RMS}}$$

(5)

$$Efficacy_{ChR2} = \frac{(I_e^{RMS} - I_{ChR2}^{RMS})}{I_e^{RMS}}$$

(6)

The RMS current used by optogenetic stimulation to suppress beta is less than that of electrical one [5]. A different mechanism of suppression is offered by optogenetic inhibition. The neuronal activity was decreased. All activity is inhibited by strong light, not just beta-activity. While not always as successful as inhibition, optogenetic excitation often requires less effective current than electrical DBS to suppress beta activity. Even in situations where optogenetic stimulation is not more successful than electrical DBS for suppressing beta-activity, the required amount of optical DBS effective current is typically near to the effective current required by electrical DBS. To summarize the thoughts, we can view this as evidence that it is more effective than electrical stimulation. Thus, optogenetic DBS can treat Parkinson’s disease symptoms more effectively due to the varied manner in which stimulations interact with network dynamics and optogenetic control of neuronal synchronization [21].

V. Comparison, Which Approach is Better?

i. Materials and methods

In the following study, some behavioral studies will compare optogenetics DBS using Chronos and ChR2 and electrical DBS.

Pre-stimulation:

Female Sprague Dawley rats were used in the following study. In optogenetic surgery, a new opsin was used that is called Chronos, as it is much faster than ChR2 and can follow high rate [100 pulses per second (pps)] stimulation [23] . In the 6-OHDA lesioned rat model of Parkinson's disease, Chronos was inserted into an adeno-associated virus serotype 5 (AAV5) with a calcium/calmodulin-dependent protein kinase II (CaMKII) regulator to affect local cell-specific expression in STN. Initially, during the surgery, the subthalamic nucleus was located using metal microelectrode recordings. The virus was then injected into the STN (DV 7.0 mm) at a rate of 0.1 l/min after waiting 5 minutes in between each 0.2-l injection (a total of 0.6 l). After the whole dosage, the needle stayed in the brain for an extra 10 minutes [23]. After that, rats received lesioning methods to induce unilateral dopaminergic neuron degeneration in the substantia nigra pars compacta (SNc), a source of the vital nigrostriatal dopamine pathway, resulting in hemi-parkinsonism. The neurotoxic 6-OHDA, which selectively destroys dopaminergic neurons (6 l, 2.5 mg/ml in 0.2 percent ascorbic acid mixed with saline, Sigma-Aldrich), has been used to imitate Parkinson's disease in experimental animals.

ii. Experiments

1. First test: Adjusting steps test

A validated indicator of parkinsonian transcripts in rats is impairments in forelimb adjustment steps, which are specifically present in hemi- parkinsonian rats. The next test involved moving each rat backward over a period of 3 to 4 seconds through a 1-m glass hallway while being kept with its rear limbs lifted. The action was captured on film, and for manual analysis, the number of steps performed with the ipsilateral and contralateral forelimbs was tallied. The delivery of continuous optical or electrical DBS was between 20 and 130 PPS DBS. Two to three trials were recorded for each session. The behavioral effects of DBS were measured by the ratio of steps taken with the contralateral forelimb to steps done with the ipsilateral forelimb. [25] [26]

2. Second test: Circling test

A single dose of methamphetamine (1.875 mg/kg in 0.9 percent saline) was administered to rats with a unilateral lesion 30 minutes before the animals were put in a cylinder to induce vigorous and prolonged circling [27] . To observe rat behavior, the cylinder was placed in a dimly lit space with an infrared camera. The presentation sequence within each block was randomly chosen to study the behavioral effects of DBS using fixed 130 pps and randomized blocks of 5 stimulation rates: 5, 20, 75, 100, and 130 pps [23]. Six trials of the fixed 130-pps laser stimulation— each consisting of 10s on and 20 s off—were used to produce the pulses, for a total of 3 minutes. The recording lasted for nine minutes. It was divided into three segments: a three-minute control phase, a three-minute stimulation period, and a three-minute light-off segment. In the rate-randomized stimulation, pulses were given for 10 seconds at each rate, followed by a 20-second off period. Each session consisted of a 2-min control period, a 2-min period of stimulation, and a 2-min time of light-off. Over time, the positions of the nose and tail's base were calculated. The data was used to manually calculate the linear speed and angular velocity. To get the normalized angular velocity or linear speed for each trial, the average angular velocity or linear speed during the pre-stimulation-off and post- stimulation-off periods immediately before and after the stimulation-on time was divided by the angular velocity or linear speed during the stimulation period [23]. Average angular or linear velocities for each stimulation rate were calculated by averaging all randomized blocks.

iii. Results

VI. Future Research

It is important to address a few problems that were not in the previous studies and research. We will declare these problems here and introduce some solutions for them:

1. Not enough / inadequate data: a major problem with regards to the research done on treating Parkinson’s with optogenetics till now is the eminent lack of experiments on real subjects. Most studies on this subject have been conducted on either network models or rats. This issue is because there are no suitable tools for the experiments to be done on humans. Another reason for this issue is that it includes some risks for the human brain. Risks include inflammation, build-up heat and implant rejection.
2. Optogenetics losing against electrical DBS in some tests: the findings emphasize the significance of considering the kinetic limits of opsins when examining the results and mechanisms of Opto- DBS. Additionally, because optogenetic responses depend on opsin expression, raising opsin expression levels may improve behavioral outcomes by enhancing optogenetic excitability. Therefore, insufficient opsin expression and a small light illumination area may cause the difference between optogenetic and electrical effects on the tests.

Possible improvements

• Another suggestion is monitoring inflammatory biomarkers for signs of any immune response. Also, a safety element can be made to check for any leakage and temperature of the brain inside the implant.
• A possible improvement for Opto-DBS is discovering a new kind of opsins. These opsins should have higher kinetic limits. It should also have a higher opsin expression level to enhance optogenetics excitability.

VII. Conclusion

Optogenetics has been in the spotlight in science for the last decade. It has been remarkably used in the study of neuroscience. It improved the therapeutic treatments of many diseases, such as Parkinson’s. Millions of people have been diagnosed with Parkinson’s, yet there is no cure for such a disease. However, its symptoms have been treated differently for many years. Still, one treatment that showed greater progress was deep mind stimulation (DBS), a neurosurgical procedure in which the brain is electrically stimulated using implanted electrodes. Optogenetics. On the other hand, improved this kind of treatment treats Parkinson’s disease symptoms by using light and opsin instead of electricity and electrodes is significantly better than electrical DBS, as it does not affect any nearby neurons that are not targeted which eliminates any side effects made by electric DBS. Opto-DBS and electrical DBS have been compared computationally by calculating the minimal photocurrent that can suppress the beta activity below the threshold. The minimal photocurrent was obtained by the Opto-DBS, which makes it more efficient than the electrical DBS. Additionally, an experiment on mice has been made to compare them by two tests: the circling test and the adjusting steps test. In this test electrical, DBS and Opto-DBS with two different opsins have been compared. However, electrical DBS showed more success in both of the tests, while only one type of opsin showed improvement. It has been concluded that each opsin has its own kinetic limits, and using a stronger opsin may enhance the effect of Opto-DBS. In our research journey, we found problems and we recommended solutions for them to increase the efficiency of the Opto-DBS. Now, one can envision a switch that can treat Parkinson's disease, but for this to come true, more research has to be done.

I. Introduction

Chronic diseases are illnesses that persist a year or longer and need ongoing medical aid, interfere with daily activities, or both. Chronic diseases like diabetes, cancer, and cardiopathy are today Egypt's main causes of death. they're thought to be accountable for 82 percent of all deaths and 67 percent of premature deaths in Egypt. Its therapy costs quite $30 trillion over the subsequent 20 years. Stem cell therapy can be a long-term cure for chronic disorders. Stem cells are undifferentiated cells with the power to duplicate endlessly (self-renewal), usually from one cell (clonal), and differentiate into a range of cell and tissue types. The utilization of stem cells to treat or prevent a disease or condition is understood as stem-cell therapy. As of 2016, hematopoietic stem cell transplantation is the only proven stem cell therapy. The only established therapy using somatic cells is hematopoietic stem cell transplantation. This usually takes the shape of bone-marrow transplantation. The sole due to avail stem cells is by building stem cell banks. The emerging demands of stem cell research and therapeutics necessitate the establishment and cooperation of centralized biobanks on a transnational and even global scale.

II. Identify, Research, and Collect Ideas

Blood is drawn from the umbilical vein before the placenta is born (in utero) or after the placenta is delivered (postpartum) (ex utero). Both systems have benefits and drawbacks. Both techniques are used at Canadian public cord blood banks, though most public banks in the United States and many European countries prefer the in-utero technique because it can be done in the delivery room by birth unit staff, is simple to learn, and does not usually require additional personnel or resources. In utero collection, strategies are used by all private banks. In comparison to the ex utero technique, comparative studies show that the in-utero procedure delivers somewhat greater volumes of cord blood and yields of total nucleated cells. Stem cells can build every tissue within the anatomy, and hence have great potential for future therapeutic uses in tissue regeneration and repair. For cells to be identified as “stem cells,” they need to display two essential characteristics. First, stem cells must have the power of unlimited self-renewal to supply progeny identical because of the originating cell. This trait is additionally true of cancer cells that divide in an uncontrolled manner whereas vegetative cell division is extremely regulated. Therefore, it's important to notice the extra requirement for stem cells; they have to be able to bring about a specialized cell type that becomes a part of the healthy animal. Many various kinds of stem cells come from different places within the body or are formed at different times in our lives. These include embryonic stem cells that exist only at the earliest stages of development and various forms of tissue-specific (or adult) stem cells that appear during fetal development and remain in our bodies throughout life. Somatic cell treatments include new technologies and therapies that aim to switch damaged tissues and cells to treat disease or injury. Stem cells have the ability to congregate in these damaged areas and generate new cells and tissues by performing a repair and renewal process, restoring functionality. ESC, iPS, and adult vegetative cell therapies, which include bone marrow stem cells and peripheral stem cells are currently being investigated or want to treat a variety of diseases. Bone marrow stem cells are accustomed to replacing blood cells in people laid low with leukemia and other cancers. Burn victims are taking advantage of somatic cell therapy, which allows for brand spanking new skin cells to be grafted as a replacement for those who are damaged. But Challenges face in hematopoietic vegetative cell transplantation in Egypt. Transplanting stem cells to every patient who qualifies in Egypt's population is expected to exceed 100 million by 2020. There are fifteen transplant centers, and the transplant rate per million is 8.4, which is significantly lower than Western standards of 36–40 per million. Until the late 1980s, when peripheral blood stem cells (PBSCs) were collected, the only source of stem cells was bone marrow harvesting. Donors' availability Patients with siblings are likely to have an HLA identical donor in the range of 25–30 percent. As a result of the increased size of Egyptian households, this figure approximates 40% of the population. However, 3% of donors registered in worldwide registries are of oriental ancestry, complicating the process of locating compatible donors for the transplant.

III. Studies and Findings

Due to the aforementioned reasons, stem-cell therapy isn't commonly employed in Egypt, despite the very fact that it's an efficient treatment for the spread of chronic diseases and injuries. The provision of vegetative cells and therefore the hazards of stem cell therapy will be derived from those facts, which are: The risks of somatic cell therapy, due to the restricted number of searches done on them. Stem cell banking is one choice to address these obstacles. The method of extracting valuable stem cells from a person's body, processing them, and storing them for future use in vegetative cell treatments is understood as somatic cell banking. Low temperatures are employed in vegetative cell banks to preserve biological characteristics and protect stem cells against contamination and degeneration. Any vegetative cell bank must use standardized and quality-controlled preservation processes to stay the cells alive for extended periods of time without losing their qualities. The most suitable declaration for most of the problems facing stem-cell therapy in Egypt is to determine a somatic cell bank.

IV. Use of Cord Blood from a Family Member Indications:

The utilization of related allogeneic transplantation using umbilical cord blood maintained in private family banks or through a directed donation program with a public bank was examined in a recent analysis of data from the CIBMTR. When bone marrow or peripheral blood stem cells from a sibling are difficult to get, such as when siblings are babies, this method may be effective. Between 2000 and 2012, the CIBMTR received reports on 244 patients from 73 different centers. Acute leukemia (37 percent), thalassemia or sickle cell disease (29 percent), Fanconi anemia (7 percent), and genetic red cell, immunological, or metabolic problems were the most common reasons for transplants (18 percent). More than 500 patients have been transplanted, according to the Eurocord Registry. The majority of the recipients were kids, and all but 29 were HLA-matched. Patients and their families who fly abroad for related cord blood transplantation may be subjected to a greater risk of complications, which could threaten their safety and be connected with a large financial outlay. In other jurisdictions,governmental control of transplantation is vastly different from that in Canada, and this type of medical tourism is strongly discouraged for patient safety. Embryonic stem cells are pluripotent, which suggests they'll produce any or all of the various cell types found within the body. They're discovered some days after fertilization at the blastocyst stage of embryonic development, particularly within the embryoblast cell mass. These cells may well be retrieved and kept in an exceedingly vegetative cell bank following early miscarriages. The bank may also preserve duct blood stem cells. As Stem cells pullulate with the canal fluid, the fluid is easy to gather and contains 10 times the quantity of stem cells seen within the bone marrow. Each new baby's fetal membrane stem cells would be retained in his account for future needs, and a stem cells account could also be created within the stem cells bank. There is a range of additional vegetative cell sources that would be kept within the bank. The organ systems of pigs and humans are 80-90 percent identical. Pigs and humans have a surprising number of features in common. We've got hairless skin, a dense layer of subcutaneous fat, light-colored eyes, prominent noses, and thick eyelashes, for example. Due to their compatibility with the physical structure, pigskin tissues and heart valves are employed in medicine. Bhanu Telugu and co-inventor Chi-Hun Park of the University of Maryland (UMD) Department of Animal and Avian Sciences show for the primary time in a very new paper published in stem cell Reports that newly established stem cells from pigs, when injected into embryos, contributed to the event of only the organ of interest (the embryonic gut and liver), laying the groundwork for somatic cell therapeutics and organ transplantation. It's feasible that pig embryonic stem cells could be transplanted.

V. Conclusion

There would be a reliable source of stem cells if this could be accomplished. Building such a large source of stem cells would improve their availability for research, as more research on stem cells and a thorough understanding of their qualities would raise the rate at which stem cell treatment would be applied. This bank will also provide stem cells for the long-term treatment of chronic diseases, eliminating the need for frequent donations.

VI. ACKNOWLEDGMENT

Thanks to Allah and the people who helped us to accomplish all of this work, we appreciate Mrs. Gihan Mohamed for her scientific comments and discussions.

I. Introduction

The human brain has always been complicated enough to push scientists to question and investigate it. To achieve that, scientists used quantum physics. Many physicists have published on the quantum measurement problem and its relation to the observer or consciousness; it is generally done by physicists who understand deep mathematical formulae. This mystery has been resolved as quantum neurobiology refers to a narrow field of the operation of quantum physics in the nervous system, such as the emergence of higher cognitive functions like consciousness, memory, internal experiences, and the processes of choice and decision-making. However, quantum neurobiology is a field that neurologists can conceptualize more easily. Because it is based on understanding the extent to which quantum physics contributes to the higher consciousness functions of the brain. Such as the place of memory storage and recall in the biology of the brain, free will, decision making, consciousness, and different states of consciousness, and how anesthesia temporarily suspends consciousness. The human brain contains three types of memory: sensory, short-term, and long-term. Sensory memory picks just information from surroundings and stays just a few seconds. Short-term memory refers to a piece of information processed in a brief amount of time. Long-term memory enables us to keep information that can be recovered consciously (explicit memory) or unconsciously for extended periods (implicit memory). Encoding, moving data from short-term to long-term memory, is a painful process as a person should visit a specific piece of information often to signal to the brain the importance of this piece of information. Scientists have developed many techniques to trace this process, like repeating, chunking, mnemonic devices, and spacing. Although, encoding is still an arduous mission as these methods often demand many struggles and a protracted time. To handle this challenge, humans should know how their memories work and their relationships. Until now, our knowledge about the whole brain is still poor and misses lots of its behaviors as they are yet unexplainable. As mentioned, the contribution of quantum physics to quantum neurobiology is very promising as it can count to understanding the brain more clearly. Furthermore, it would present a much better idea about how these memories are connected and, in turn, what people should do to move the information that seems necessary to the long-term memory to be retrieved easier. Other research papers use quantum information science methods to model cognitive processes such as perception, memory, and decision-making without considering whether quantum effects operate in the brain [1]. Another prior study focused on the less contentious topics of quantum-related activity in the brain via quantum events [2] and super-determinism [3]. Super-determinism explains quantum phenomena in physics regardless of their physical origin and why biological systems of sufficient complexity display quantum-like behavior.

Human Brain

It is already known that quantum mechanics plays a role in the brain since it determines the shapes and properties of molecules like neurotransmitters and proteins, and these molecules affect how the brain works. Consequently, drugs such as morphine affect consciousness.

II. Quantum Neurobiology

These cutting-edge developments in neurobiology pave the way for quantum neurobiology and promote whole-brain neuroscience research goals such as full-volume, three-dimensional analysis of the whole brain at numerous spatial and temporal dimensions. The urgent practical problem is to integrate data from EEG, MEG, fMRI, and diffusion tractography (nerve tract data) [12]. Quantum techniques are required because supercomputing (which can only represent one-third of the human brain in a recent experiment [13]) and other conventionally based technologies demonstrate that new platforms are required for the future. The following stages of neuroscience data analysis Simultaneously, quantum information technology is establishing itself as a significantly more scalable platform with three-dimensional modeling capabilities suitable for representing real-life brain processes such as neurons, glia, and dendritic arbors. Quantum approaches enable the investigation of new classes of neurobiological problems, such as neural signaling with synaptic integration (aggregating thousands of incoming spikes from dendrites and other neurons) and electrical-chemical signal transduction (incorporating neuron-glia interactions at the molecular scale). This work describes the three areas of activity developing in quantum neurobiology (Table 2). It proposes a novel theory of neural signaling (AdS/Brain, based on the AdS/CFT correspondence (anti-de Sitter space/conformal field theory)).

III. Quantum-Aided Scanning Application

i. Wavefunctions:

Interpreting empirical data from diverse brain scanning modalities using wavefunctions, a mathematical representation of an isolated quantum system state, and quantum machine learning is the first broadly used class of quantum neurobiology applications. Since 1875, researchers have studied the scalp's EEG-detectable potentials [14] , but a more comprehensive understanding of brain signaling also considers waveforms associated with astrocyte calcium signaling, neurotransmitter activity, and dendritic spikes [15]. The intractability of the Schrödinger wave equation has generally required that EEG data be interpreted with efficient nonlinear wave models, even though quantum mechanical wavefunctions are readily suggested. This work is being replaced by quantum algorithms, which are being utilized to recreate medical pictures from MRI, CT, and PET scanners. Applications for quantum BCIs (brain-computer interfaces) that interpret EEG waveform data in a network of brain-machine communication may become available soon [16].

ii. Quantum EEG:

Finding the optimal wave function to match the massive amounts of EEG data created requires quantum machine learning, which is quickly becoming an essential technology. Classifying Parkinson's disease patients' EEG data as possible candidates for Deep Brain Stimulation by examining 794 features from each of the 21 EEG channels is a usual challenge. The use of machine learning techniques in a quantum environment, formulating classical data using quantum approaches, and investigating quantum problems using machine learning techniques are all examples of quantum machine learning. The three primary machine learning architectures—neural networks, tensor networks, and kernel learning [17]—all have quantum formulations available. A quantum perceptron has been created for presently accessible quantum processors (the IBM Q-5 Tenerife). In addition, a quantum recurrent neural network (RNN) and a quantum convolutional neural network (CNN) with a more sequentially oriented quantum circuit topology has been proposed to model EEG wavefunctions using quantum neural networks. The 10-20 System of Electrode Placement is a method used to describe the location of scalp electrodes. These scalp electrodes are used to record the electroencephalogram (EEG) using a machine called an electroencephalograph. The EEG is a record of brain activity and that which was used in our experiment. Electrode placement using the Extended International 10-20 system (10% system) covering prefrontal (red circles), dorsolateral prefrontal (purple circles), ventrolateral prefrontal (orange), frontal (green), temporal (blue), parietal (yellow), and occipital (gray) cortices. Odd numbers = left, even numbers = right, and z = zeros in the midline (as shown in figure1). Quantum spike-activated neural networks (SNNs), a bio-inspired neuromorphic computation model with threshold-triggered activation akin to the natural neuronal firing of the brain, are an alternative to quantum machine learning. Examples of quantum SNN initiatives include accelerated matrix processing using synaptic weighting and superposition modeling and emergent behavior research using Josephson junctions [18]. EEG data interpretation is a signal processing task involving noise reduction, feature extraction, and classification exercises. For instance, the signal-to-noise ratio of EEG data used in BCIs is poor because of noise. A quantum approach uses a quantum recurrent neural network (QRNN) framework to apply filtering algorithms based on advancements in processing techniques (Kullback-Leibler spatial patterns and Bayesian learning) [19]. The Schrödinger wave equation and a Hamiltonian are used to analyze the time-varying wave packets that the QRNN uses to describe a nonstationary stochastic signal (energy operator). Real-time EEG data and BCI competition test data are used to test the QRNN and find that it performs better than conventional Kalman filtering techniques. The feature extraction and classification portion of EEG data analysis are also carried out using a variety of quantum machine learning techniques, including independent component analysis, wavelet transforms, and Fourier transforms [20], as well as entropy-based quantum support vector machines, evolutionary algorithms inspired by quantum mechanics, and quantum-inspired support vector machines. Finally, EEG data analysis is now possible at a new level of data resolution because of quantum technologies. Using standard cortical building blocks in the time and frequency domains, one study examines single-trial event-related potentials (EEG segments time-locked to cognitive events), while another route integral modelizes electrical impulses and calcium-ion interactions.

iii. Quantum Protein Folding

Both classical and quantum techniques have improved the computational complexity of protein folding, which is NP-hard. Predicting a protein's three-dimensional structure from its underlying amino acid sequence is difficult. It is believed that an accumulation of misfolded proteins is the root cause of many neurodegenerative illnesses, including Alzheimer's and Parkinson's [21]. Traditionally, an important endeavor is AlphaFold, as Google's DeepMind team has demonstrated in the CASP-14 data competition [22] by extending its success in game playing to protein folding. By focusing on global restrictions, such as available space, rather than solely local sequence interactions, an attention-based technique is employed to generate atomically exact configurations. Progress is also being made with quantum techniques, especially with quantum annealing machines that simulate protein folding as a low-energy optimization. The many amino acid sequences in the protein are represented spatially by a lattice. While research frequently focuses on neuropeptides as small protein strings that can be easily used as intervention targets, the median length of a human protein (375 amino acids) can be easily analyzed using annealers. In a study of 30,000 protein sequences using protein Hamiltonians, one lattice-based quantum protein folding experiment discovered that minor adjustments significantly increase folding efficiency [23]. A related experiment used the IBMQ Poughkeepsie 20-qubit quantum computer to show the lattice-based folding of a 7-amino acid neuropeptide. The QFold project suggests a quantum algorithm based on the torsion angles of amino acids, deployed with quantum walks (on the IBMQ Casablanca quantum processor), as an alternative to lattice structures.

IV. Neuroscience Physics

The neuroscience interpretation of results from fundamental physics is called neuroscience physics. Applications covered here include the brain Hamiltonian, a group of AdS/Neuroscience theories based on the AdS/CFT correspondence (AdS/Brain, AdS/Memory, AdS/Superconducting, and AdS/Energy), neuronal gauge theories (symmetry-breaking, energy-entropy balances), network neuroscience, and random tensors (high-dimensional systems). Neural signaling, a subject involving electrical-chemical signal transduction and synaptic integration (aggregating millions of incoming spikes from dendrites and other neurons), is of particular interest (incorporating neuron-glia interactions at the molecular scale). The multi-variable partial differential equation (PDE) functionality required to represent inter-neuronal spatial interactions is not included in the conventional compartmental models utilized in computational neuroscience [23, 24]. Diffusion-reaction equations, for instance, are one way to integrate the activity of dendritic spikes, which includes protein cascades in dendritic arbors, astrocyte calcium signaling, and the transfer of molecules using protons and ions on the quantum (atomic and subatomic) scales [26] . In order to represent brain signaling, proposed quantum neurobiological theories consider multi-scalar models, phase transition, nonlinear dynamical systems, energy-entropy relations, and high-dimensional representation.

i. AdS/Brain

The AdS/Brain theory combines the four scale tiers of the network, neuron, synapse, and molecule, and is a multiscale explanation of neurological signaling based on the AdS/CFT correspondence. With increasing levels of bulk-boundary communication, the theory represents the first illustration of a multi-tier interpretation of the AdS/CFT connection. The matrix quantum mechanics formulation (multi-dimensional matrix model [27]) with bMERA (brain) random tensor networks generated using Floquet periodicity-based neural dynamics is the suggested implementation of the AdS/Brain theory. According to the AdS/CFT correspondence (anti-de Sitter space/conformal field theory), a boundary theory can adequately describe a physical system with a bulk volume in one less dimension. According to the theory (gauge/gravity holographic duality), a gauge theory or quantum field theory's boundary surface and a gravity theory's bulk volume are equivalent in terms of dimensions. With applicability across all physics arXiv sections, the study is one of the most cited publications in any field (nearly 21,000 references as of December 2021). In the AdS/CFT correspondence, the same system is seen from two angles, and the equations for the solution are provided. The AdS/SYK (Sachdev-Ye-Kitaev) formulation begins with a known classical gravity theory (Einstein gravity) in bulk to solve an unidentified quantum field theory describing a superconducting material on the border, which is a typical bulk-to-boundary use case. For example, black holes and unusual materials can be compared mathematically because both systems share mass, temperature, and charge-related characteristics (classical gravity bulk). On the other hand, boundary-to-bulk deployments begin with a well-known quantum field theory on the boundary and try to develop an emergent structure theory, such as an unidentified quantum gravity theory, in bulk [28] . A current field of research is establishing bulk-boundary mappings, including a quantum error-correcting setup (which safeguards a logical qubit in bulk by connecting it to an ancilla of physical qubits in the boundary). The AdS/Brain theory proposes the first instance of a multi-tier correspondence (multiple graduated levels of bulk–boundary relationships) to instantiate the four scale tiers of the brain network, neuron, synapse, and molecule (and could be expanded to other tiers). The model accommodates the brain's neural signaling processes between axon, presynaptic terminal, synaptic cleft, postsynaptic density, and dendritic spiking potentials from dendrite to soma. The bulk–boundary pair relationships are network–neuron, neuron, synapse, and synapse–ion. The scales and measured signals are

• local field potentials at the brain network level (10−2 m),
• action potentials at the neuron level (10−4 m),
• dendritic spikes at the synapse level (10−6 m), and
• ion docking at the molecular level (10−10 m).

The AdS/Brain theory deals with the need for renormalization in multi-scalar systems (the ability to view a physical system at different scales). In order to account for the reality that all conceivable particle positions and occurrences are genuinely feasible, renormalization algorithms must address the infinities that arise in quantum physics. As a mathematical tool for smoothing systems to be examined at various scale tiers based on various parameters, various renormalization group (RG) methods have been presented (degrees of freedom). The multiscale entanglement renormalization ansatz (MERA), which implements an iterative coarse-graining strategy to renormalize quantum systems based on entanglement or other properties, is a significant advancement [28]. The MERA tensor network has a topology compatible with the AdS/Brain theory and is used in a bMERA (brain) implementation. It comprises alternating layers of disentanglers and isometries that combine a multi-tier system into a single perspective. Different neural dynamics paradigms determine the system evolution at each scale tier of the neural signaling operation, which is the second criterion the AdS/Brain theory addresses. The basis for a multi-scalar model of the brain network, neuron, synapse, and ion channel dynamics is Cloquet periodicity [29, 30] pushed with continuous-time quantum walks [32] , as these formalisms flexibly handle different dynamical regimes within a system.

ii. AdS/memory

The AdS/CFT correspondence is applied in neuroscience to study the issue of information storage in AdS/Memory. The study program tackles the computational neuroscience issue of memory formation using the AdS/CFT correspondence (in the form of black hole physics) [33]. Critically excited states may cause the adequate information storage found in black holes and brains. For example, a quantum neural network adds entropy scaling by area rather than volume and holographic characteristics. The critical states (neuron excitatory synaptic connections based on gravity-like interaction energy) produced by the quantum optical neural network (with qudit-based bosonic modes) have an exponentially increased capacity to retain information. The investigation of a system's capabilities in a highly excited state—instead of just determining the system's ground state and associated energy tiers—is novel. A highly excited critical state, maybe rather than the ground state, has the highest quantum memory capacity. The conception of brain signaling as a criticality-triggered phase shift may have immediate ramifications for quantum memory and quantum neurobiology. One strategy being used, for instance, to identify new matter phases in systems that cannot reach thermal equilibrium is to consider the system extremes [30].

iii. Random Tensors:

For the treatment of high-dimensional multi-scalar systems, random tensors are a tensor network technology comparable to MERA tensor networks (computation of entangled quantum systems). A structure known as a tensor network is used to represent and manipulate multi-body quantum states. It is created by factorizing high-order tensors (tensors with many indices) into a collection of low-order tensors, whose indices are added to create a network with a specific pattern of contractions. Random tensors have been tested for up to five dimensions (rank-5 tensors), generalizing random matrices (2 2 matrix formulations) to three or more dimensions [34]. In order to apply the AdS/Brain theory as a tensor field theory of neural signaling, random tensors offer an additional model (in addition to matrix mechanics). For example, on quantum platforms, existing neural field theories could be instantiated as tensor field theories [35] (using three-state neurons [36]). Similarly, the AdS/Brain theory's four dimensions (network, neuron, synapses, and molecule) could be indexed with rank-4 random tensors, modeling the transition from a quiescent to firing signal as a matrix(2d) to tensor(3+d) phase transition (planar to malonic (high-dimensional) graph representation). The sophistication of conventional computational neuroscience compartmental models is increased by the dimensionality required by these tensor field theories and malonic graphs of neural transmission to instantiate synaptic integration research findings. The dendritic response is shaped partly by spine density gradients, and reducing spine density enhances thresholded signal pooling (some neurons combine the outputs of numerous dendrites that have been individually thresholded) [37]. Dendritic geometry's curvature produces pseudo-harmonic functions that can anticipate dendrite concentrations and their possible role in signal processing, according to nonlinear models used to examine the postsynaptic density and dendritic shape as elliptical spheroids [38]. Implementations of differential geometry provide a fresh perspective on studying mitochondrial membrane architecture, whose metabolic dysfunction may be a factor in neurodegeneration [39]. As a result of their idealized geometries, conventional methods of modeling mitochondria (ATP and heat) are inadequate because they distort metabolic flux. However, using differential geometry to empirical TEM tomography data, a more reliable analytical model based on Gaussian curvature, surface area, volume, and membrane motifs are connected to the mitochondria's metabolic output and calls for a multidimensional approach—can be created. Furthermore, these differential geometry techniques may be used to treat the mitochondrial bioenergetic stress response [40].

iv. AdS/Energy (Brain Hamiltonian)

With the AdS/CFT mathematics, which renormalizes entanglement (correlations) across system levels, the AdS/Brain theory offers a generalized multi-scalar model of neuronal behavior interpretable at different bulk-boundary size tiers. The primary multi-scalar quantity is entanglement. However, energy-related formulations (written as a Hamiltonian) are also feasible. The first law of entanglement entropy (FLEE), which has been characterized as the first law of thermodynamics, states that a change in border entropy is comparable to a change in bulk energy (Hamiltonian) [41]. A formalism for resolving the AdS/CFT correspondence in terms of energy did not previously exist, although energy formulations are essential to quantum systems. With the AdS/CFT mathematics, which renormalizes entanglement (correlations) across system levels, the AdS/Brain theory offers a generalized multi-scalar model of neuronal behavior interpretable at different bulk-boundary size tiers. The principal multi-scalar quantity is entanglement; however, energy-related formulations (written as a Hamiltonian) are also feasible. The first law of entanglement entropy (FLEE), which has been characterized as the first law of thermodynamics, states that a change in border entropy is comparable to a change in bulk energy (Hamiltonian) [41] . A formalism for resolving the AdS/CFT correspondence in terms of energy did not previously exist, even though energy formulations are essential to quantum systems.

V. Methodology & Purpose

When charged, what physical information is received by the brain and transduced or transmitted by neurons? What is the neurotransmitter's endpoint, and what form does it take in the brain? In other words, what exactly is the nature of "memory"? Positions are also employed. The American Electroencephalographic Society established the locations and nomenclature of these electrodes. P10, which had 1, 3, 5, 7, 9, and 11 for the left hemisphere, representing 10%, 20%, 30%, 40%, 50%, and 60% of the Inion-to-Nasion distance, respectively. with the understanding that the new letters are not necessarily associated with the area beneath the cerebral cortex. For 36 and 72, electrodes were placed alternately on the scalps of three normal-life subjects, not patients, at F, T, C, P, and O, including "A" letter positions on middle line locations. The participants ranged in age from 18 to 22 years. Later, a younger age group of 25 to 35 years old was recruited for the EEG recordings. These recordings were made in July and August of 2011. Unfortunately, the 72-electrode data had to be rejected due to improper electrode placement and an excessive number of artifacts that could not be adequately removed. Fortunately, earlier recordings were available for research. This experiment tested subjects' recollection of three states: deep sleep, in-between waking, deep sleep, and wakefulness. Their recollections and responses to questions were compared and analyzed retrospectively with Delta, Theta, Alpha, Beta, and Gamma waves observed in the above recordings to better understand individuals' vis-a-vis memory and brain frequencies. Questioning on recollection of condition in a deep sleep and before and after waking up confirms that the "Self-induced" signal is indeed related to the old term "ego" and that I exist. Denials are also included. The self-awareness brainwave signals are active at frequencies above 5 Hz but not below. Previously, frequencies ranging from 0 to 4 Hz had a "witness" function, allowing an individual to recollect and recount events. However, the individual is in a deep sleep at 0 to 4 Hz and never narrates that condition. Readers can use introspection to put this to the test. Why don't we remember, based on this research and questioning? What is our state in a deep sleep, and why do we remember a few things? Dream Visuals versus Others? A self-identifying signal, an objective frequency of self-awareness, must exist. Unfortunately, that objective signal is absent during deep sleep.

VI. Results

One remarkable finding of brain wave patterns is frequencies ranging from 0 to 40 hertz. Internally the sent signal referred to as "self-induced" contains pulse energy. Propagation begins at 5 Hz, 8-12 Hz, 20-40 Hz, and higher. They emerge in fully awake settings at frequencies ranging from 0 to 12Hz to 40Hz. The "self-induced" data signals contain material connected to "I," "I exist," which is a physical self-identity signal, as well as denials, such as "I don't know," "I don't see," and so on. "I" stands for "Self-Awareness," even though "I" is a human-made auditory signal within a language. Self-awareness brainwave waves are active at frequencies of 5Hz and higher. Hundreds of sounds in the world's languages correlate to the letter "I." Sensitivity to the universe of information is caused by receptor neurons' conversion of this signal into those generated signals. According to the above assertion, the "self-awareness" signal must convert farther from the 5 Hz frequency. The reported 5 Hz EEG "signature" of the "I", is supposed to be universally accepted among neurologists and clinicians as a part of the theta rhythm (5-7.5 Hz). Recorded from the frontal and temporal regions of the brain and commonly found in children with behavioral disorders or, in general, pathological states of the brain, the presented result does correspond to normal human brain function relating to dream state and self-awareness signals with 5-8Hz frequency, which is recollected as the faint visuals perceived because of very low energies of 2.0678e-14 eV to 3.3085e-14 eV. All human brain activity has 5-7Hz rhythms and continues to increase. This low-frequency activity does not imply that all humans suffer from pathological or behavioral issues. Because it is not feasible to get deep into a living brain to study the source of the brain or mental activity, a simple analogy is taken from a movie screen mechanism. The images of the physical world and the characters are nothing more than light rays displayed on the screen. They are the light frequencies acquired on film frames during the shooting process. Light from the projector flows through the film frames and converts, according to a matrix of dots, into the light frequencies obtained during filming, which subsequently show as pictures and action on the entirety of the covering screen. Similarly, on a Compact Disc, data produced by laser light is stored in a series of minor dents and planes (called "pits and lands") and imprinted in a spiral data track onto the top of the polycarbonate layer. An infrared semiconductor laser beam with a wavelength of 780 nm is used to read the programmed information through a lens at the bottom of the polycarbonate layer. The reflected laser beam off a CD's "pits and lands" is translated into audio and visual signals of laser beam strengths at different frequencies matching to the "pits" dimension and staying original when reflecting off the "lands." The self-awareness signal does flow via an infinitesimal gap or hole inside the atomic structure, shifting the frequency of self-awareness to the frequency of received energy, resulting in this gap. The frequency of self-awareness corresponds to the dimension of the gap or hole in the atomic structure. In other words, the self-signal becomes the light signal reflected by the previously observed object. As a result of this conversion and reversal to self, the individual feels they have a recollection of the item. Normal brain function is the conversion of self-awareness into frequencies of objects and sounds perceived millions of times. When this activity is hyper, and the self-aware signal is not coming back or does not reverse, the individual's mental health is disturbed. Such a condition of loss of self-awareness creates health and behavioral problems. Is the world around us sending us any information about its natural state? There is no projector, no light, no film to detect external light, and no screen to show the image of the physical world in the brain. There is also no mechanism for recording and reading a compact disc. Nonetheless, the registered light When frequency coding in the nucleus of lateral geniculation is reactivated, a weak image of the seen environment is projected in the visual cortex or primary visual cortex (V1). The projected image on the movie screen and in the brain corresponds to the light reflected off the bodies. In other words, reflected light is perceived in visual perception. The physical universe, including humans, contains no information. Indeed, the reflected light has no physical, physiological, chemical, biological, Body molecular, or atomic information. The original frequency of light is modified, effectively attenuating, and the color attribute has changed frequency at the time of impingement and reflection (at light speed). Light has two properties: color and luminosity. Even light from self-luminous sources such as the Sun and stars does not contain information about the matter makeup of such things. Neither are "physical bodies" on the screen nor in the brain. In essence, the claimed recollection of the physical world is a self-imposed "False Memory." This "false memory," kept strongly or compulsively in the brain, causes conflict and disrupts mental circumstances. It can be inferred that this memory, for practical reasons embedded in the day-to-day lives of individuals, helps organize life. Memory reactivations from 5Hz up to 12 Hz appear between wake-sleep states (when an individual is neither fully awake nor in a deep sleep). The narration of images, called a dream, is of different intensities; hence the individual can sometimes narrate those images clearly, and at other times he or she cannot recollect the images. The above two states of dream images correspond to high and low intensities of brain frequencies. Between 8 Hz and 12 Hz, brain waves carry a certain intensity of image resolution, which the individual then recollects and narrates. The low intensity of image resolution, which appears between 5 and 8 Hz, for example, 7 Hz = 2.8950e-14 eV, of brain frequency, is not clearly remembered. Therefore, the individual may express indistinct recollections of some images representing obscure visuals manifested just after a deep sleep. In other cases, the frequencies are close to the waking state because the intensity is higher, 10 Hz (9.671 957 Hz = 4.1357e-14 eV, implying the possibility of remembrance [19, 19a] . In a few other cases, due to higher frequency activity, between 8 and 12 Hz, having an energy content of 3.3085e-14eV to 4.9628e-14eV, individuals experience an admixture of visual data which creates a non-cohesive image display or dream sequence. The energetic activity corresponding to induced signals by sense perception is, in fact, consciousness active in energetic form. In other words, "Active Consciousness is Energy."

VII. Conclusion

All biochemical, bio-physiological, and biophysical components of the human brain become inactive after death, as was previously discussed. The key to unlocking the riddle of memory is a living brain, which is nothing more than an energetic activity. The epistemological study of pure-natural physics and the foundations of sense perception are presented to comprehend the basis of memory. A person's waking, dreaming, and deep sleep electroencephalography (EEG) signal data are also analyzed. The study investigates whether the human brain is aware of the inherent order of the physical universe. The current paper's main concern, putting all of this into perspective and utilizing science to get the results, is the application of quantum information science to problems in neuroscience, which is concerned with potential quantum effects occurring in the brain. The study explores a new generation of quantum technologies inspired by biology In the emerging field of quantum neurobiology. The first group uses wavefunctions and quantum machine learning to analyze empirical data from genetics, protein folding, and neuroimaging modalities (EEG, MRI, CT, and PET scans). Those that establish brain dynamics as a comprehensive framework for quantum neurobiology, including superpositioned data modeling assessed with quantum probability, neural field theories, filamentary signaling, and quantum nanoscience, come in second. The third category is the interpretation of fundamental physics results in the context of neurobiology by neuroscience physics.

I. Introduction

The sun is one of the main reasons that gives us the ability to live on the earth. In astronomy, many questions about the sun have been asked. Like, what is the sun made of? How does it produce that enormous amount of energy over this long period of time? Why will it disappear one day? Fortunately, most of these questions have been asked while studying the sun. Many techniques and discoveries have made humanity reach this level of knowledge about the sun. For example, studying the emission spectrum of the sunlight made the elements of the atmosphere of the sun known. Also, adding specific filters to the telescopes helped in discovering many phenomena of the sun’s surface like the sunspots, the active areas, and the solar winds. It is very hard to investigate the interior of the sun with astronomical tools; moreover, it is even impossible to send a spacecraft to gather information because of its high temperature (around 15 million degrees) [1]. So, the best method was to make theories about what happens at the interior of the sun like the thermonuclear reactions that provide us with energy. But how did astronomers prove that these theories are true? The key to answer this question is to investigate more about the nature of neutrinos.

II. Thoughts About The Sun

Some of the old thoughts about the sun are discussed below.

i. Ancient Civilization Thoughts

The role of the sun as an energy source and one of the main reasons for the continuous life on earth has been well known since the start of humanity. The sun has played many roles in different civilizations over time. It was considered as a god in some civilizations like in the ancient Egyptians (the god Ra) and ancient Greeks (the god Helios). The Mayans built the pyramid of Kukulkan in El Castillo. its axes run through the northwest and southwest corners of the pyramid and are oriented toward the rising point of the sun at summer solstice and its setting point at the winter solstice. In Chaco Canyon, there were several structures that indicate their understanding of the sun’s movements. For example, the special corner windows in Pueblo Bonito let light in, but only as the days get closer to the winter or summer solstices. During the summer solstice, a window on the south wall of Casa Rinconada allows a beam of light to enter a niche on the back wall. At the equinoxes and solstices, Fajada Bute casts a brilliant ”Sun Dagger” in one or occasionally two slender shafts of light frame a spiral petroglyph [1]. The first attempt to scientifically study the energy source of the sun was not until the middle of the nineteenth century.

ii. Early ideas about the source of the sun’s energy

There were some thoughts that the sun’s energy comes from the combustion process (burning fossil fuels like coal and natural gas is an example of such a process). These thoughts can also be be easily falsified using the following calculations: The amount of energy that is released from burning is $10^{−19}$ joules per atom. Knowing that the luminosity of the sun is approximately $3.9 \times 10^{26}$ joules per second, these burning processes would consume $\frac{3.9 \times 10^{26} joules \; per \; second}{10^{-19} joules \; per \; atom} = 3.9 \times 10^{45} atoms \; per \; second$. Since the Sun contains about $10^{57}$ atoms, the time required to consume the entire sun by burning is $\frac{10^{57} \; atoms}{3.9 \times 10^{45} \; atoms \; per \; second} = 3 \times 10^{11} \; seconds$. This period of time, which is about 10,000 years, is shorter than the actual age of the earth [2, Ch. 16, p. 405].

iii. Kelvin-Helmholtz contraction

In the mid-1800s, the first attempt for humanity to find out the energy source of the sun was made by the English physicist “Lord Kelvin” and the German scientist “Hermann von Helmholtz”. They thought that the huge mass of the outer layers of the sun was compressing the inner gas layers in a process called Kelvin-Helmholtz contraction, which would result in increasing the temperature of these gases (this is similar to what happens when you pump air into a wire, as the pressure of the coming air increases, you can notice an increase in the temperature of the wire). According to the ideal gas law, this theory would make sense, and it happens to take place at the early ages of star formation. But, for this theory to take place, the sun would have been much larger in the past, which is not true. Also, Helmholtz’s own calculations showed that if the sun would have started its initial collapse from a solar nebula (a very wide range of gases and dusts that normally collapse on each other forming stars), it would happen from no more than 25 million years ago (recent evidence shows that the sun have at least lived for more than 4.56 billion years). So, this theory was proven wrong [2, Ch. 16, p. 405] [3].

III. The Thermonuclear Theory

The newest and most convenient theory about sun’s energy source is the thermonuclear theory. This theory states that in order for the sun to emit such huge amounts of energy, it must undergo some nuclear reactions. Such reactions cannot happen unless a very huge temperature and pressure was met, like what happen at the core of the sun with a temperature of about 15 million degrees kelvin and a pressure of about $3 \times 10^{16}$ pascals [4] (this corresponds to a pressure 300 billion times of the earth atmospheric pressure). In such conditions, the protons are moving freely with no electrons pounding with a very high speed (nearly 500 km/s) that allows them to overcome the repulsion between each other.

i. The p-p chain

ii. Proving the thermonuclear theory

For the scientists to prove that this process occurs at the core of the sun, they have two options:

1. Analysing the gamma rays ($\gamma$) coming from the sun.
2. Proving the existence of neutrinos coming from the sun.

IV. Neutrinos

Neutrinos are neutral, nearly massless particles, which are produced by the thermonuclear reactions happening in the sun. They have a huge abundance in the universe. Neutrinos were theorised for the first time in 1930 by Wolfgang Pauli as a way to balance out the energy in a reaction called beta decay, moreover, beta decay is a radioactive decay process that releases beta rays from an atomic nucleus. The proton in the nucleus changes from a proton to a neutron during beta decay, and vice versa. [7]. Neutrinos are playing a fundamental rule of making up everything in the universe. They have many abilities that have made them worth being studied, and very helpful in many astronomical discoveries. So, in order for the scientist to detect the neutrinos coming from the sun, they have to understand their properties.

i. Detecting Neutrinos

Fortunately, in the 1980s, the direction issue was fixed after the Japanese experiment Kamiokande made by the physicist Masatoshi Koshiba. In this experiment, instead of using perchloroethylene ($C_2C_{14}$), he decided to use 3000 tons of water ($H_2O$) surrounded by 1100 light detectors. So, if a high-energy neutrino hit one electron of the water molecules, it would produce a streaking flash of light. After that, this light flash will be collected by the light detectors and analyzed on the computer. As expected, after collecting neutrinos and analysing data, the results had shown that the collected neutrinos in Raymond’s experiment were solar neutrinos. It was a great thing that it finally proved that the sun is making thermonuclear reactions at its core, but the result of the Kamiokande experiment had not given any explanations about the 2/3 of the solar neutrinos that were not detected [8] .

This leads us to the next experiment, which was the Canadian Sudbury Neutrino Observatory (SNO) device. This device used the same method as in the Kamiokande experiment, except it used heavy water instead of light water. In the molecule of the heavy water, the hydrogen nucleus is composed of one proton and one neutron (the same as the $^2H$ nucleus mentioned in section 2.2), when a high-energy neutrino of any type reacts with this nucleus, it will result in kicking out the neutron to be observed by another nucleus and emit energy. This energy will be collected using light detectors as in the Kamiokande experiment. The results of this experiment have been roughly the same as the expected number of neutrinos coming from the sun. Thus, there were not any “Solar neutrino problem”, the problem was in the lack of the detecting techniques of different types of neutrinos [2, Ch. 16, p. 413], [9].

ii. Properties of The Neutrino

As a by-product of the nuclear reactions, the neutrino has a very little mass (recent evidence shows the value of it to be $1.25 \times 10^{-37} \, km$). To imagine how light this is, you need to combine about 7.2 million neutrinos to get the mass of a single electron. Thus, due to its very tiny mass, the neutrino can travel with a very high speed (near the speed of light). Another important property of the neutrinos is their ability to interact with nearly no matter. They have advantages over the gamma rays. Neutrinos immediately leave the core of the sun after they are produced and they reach the earth in a very short period. On earth, there are approximately $10^{14}$ solar neutrinos (neutrinos that are coming from the sun.) passing through each square metre of its surface each second [2, Ch. 16, p. 412] but the challenging part lies in detecting them as they nearly do not interfere with matter.

iii. Types of Neutrinos

It was thought that there was just a single type of neutrinos until in 1968, but it was shown that there are three major types of neutrinos, which are the electron, muon, and tau neutrinos. (they have been named after the particles they came from.) [10]. Actually, scientists have never seen neutrinos before, they only have seen the other particles produced when the neutrino interacts in a detector. In 1998, it was found that a neutrino can change its type according to its way of oscillating. This happens to the solar neutrinos on their way from the core of the sun to the earth.

iv. Neutrino Energy

V. Antineutrino

Antineutrino is the antiparticle of the neutrino, which means that they share the same mass but have opposite signs. The fact that neutrinos and antineutrinos both are neutral do not mean that they are the same. In fact, they differ in something that is called the “lepton number.” Before diving deep into the meaning of the lepton number, let’s first talk about the types of antineutrinos and how each type is created [12].

i. Types of Antineutrinos

As its antiparticle, the antineutrino has three major types. The electron antineutrino, the muon antineutrino, and the tau antineutrino. The electron antineutrino was firstly discovered in the decay of the neutron. As a result of the neutron decay, it should be converted to a proton and an electron. But while observing the electron resulted from such a decay, it was noticed that its energy is less than the predicted energy from the law of conservation of energy, which means that the decay of the neutron has an emission of a third particle, which is the electron anti neutrino [13].

ii. Lepton Family and The Conservation of Lepton Number

Leptons are said to be elementary particles, which means that they are not composed of any smaller units of matter. This family is divided into three main categories:

1. Electron neutrino
2. Muon neutrino
3. Tau neutrino

Every particle in this family is either negatively charged or neutral. For the electron, tau, and the muon, they are negatively charged. For the muon neutrino, the tau neutrino, and the electron neutrino, they are neutral. Each one of these elements has its distinct mass. The electron has mass of 1/1,840 that of the proton. The muon is much heavier with a mass equal to 200 electrons, and the heavier one ,tau, with a mass of 3,700 electrons. The combined mass of their neutrinos is less than 1/1,000,000 of the electron mass.

VI. Benefits Of Studying The Neutrinos In Geology And Communication

In this section, some of the uses and benefits humanity will gain by studying the neutrino and its antiparticle is discussed. Most of these inventions and uses are still under research, and they will not be available before a couple of years of studying the nature of these particles.

i. The Neutrino Geology

This project aims to detect the underground energy sources and other economical minerals (like gold, iron oxide, and copper). The essence of this project is to detect the geo-neutrinos coming from the beta decay in the earth’s crust. There are around 20 beta-decay long-lived in the earth’s crust. But the only important four are: Uranium ($^{238}U$), Uranium ($^{235}U$), potassium ($^{40}K$), and Tor ($^{232}Th$). The reason for detecting such decay reactions is that most of the geological structure from which minerals are extracted are distinguished by an increased U/Th content. For example, an increase in the U/Th content would mostly result in high TOC (Total Organic Compound) level. It is also possible to estimate where there are oil or natural gas deposits in each location. The research phase of this project has began in October 2019. It is divided into three main steps The first step is to research and develop a prototype of a 5-kilogram neutral particle detector, test and calibrate it with the neutrinos in the laboratory conditions at NCBJ Świerk (National Center for Nuclear Research). The next step involves the development of a prototype 50-kilogram neutron and neutrino detector. In addition to a 500- kilogram neutrino detector. For the third and final step, the development of a modular system of portable geo-neutrino detectors consisting of several 500-kilograms devices cooperating with each other is involved. Fortunately, recent evidence has shown the possibility of developing a detector with a significant increase in the cross-section reacting with the neutrinos with a factor of 10,000. This would result in decreasing the mass of the detector from 1,000 tons to about 1 ton, which would result in creating mobile detectors. This project is still in the research phase, but once it is completed it will result in a huge transformation in the exploration for energy sources [14].

ii. Using Neutrino In Communication

VII. Conclusion

Since the start of history, humanity’s curiosity and thirst for knowledge have led to many discoveries and inventions. Its love of development and satisfaction of knowing the reason for each phenomenon had made us move from the sacred thoughts about the sun, into scientific thoughts that aimed to discover the energy source of the sun. However, in this paper, the falseness of these old thoughts has been proven. In addition to proving the thermonuclear theory; however, doing that has spot the light into the importance of the neutrino and its family to the past and future of humanity. Starting from its role in proving the thermonuclear theory, explaining it and its antiparticle, and making proposals for its future uses like communication, exploration, and many other uses. Hoping that the neutrino will be one of the essential parts of life in the future, since J. J. Thomson did not know that his discovery of electrons, after 100 years, would be one of the most essential things in our life.

I. Introduction

The squirrel can survive any fall from any height without having a serious injury. That ability helps it in its high-altitude habitat. It can obtain those results by manipulating certain factors, such as drag force. In fluid dynamics, drag (fluid resistance) is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid [1]. This can exist between two fluid layers (or surfaces) or a fluid and a solid surface. In this paper, the latter one will be considered. Furthermore, the fluid will be air, and the solid surface will be the squirrel: the squirrel will be gliding through air. Drag force depends on many factors including density of the fluid, speed of the object relative to the fluid, drag coefficient, and cross-sectional area. Considering all the factors mentioned, when the drag force equals the weight, the resultant force equals zero, which results in zero acceleration based on second law of Newtonian dynamics (Force = mass * acceleration). Thus, the object moves with constant velocity. That constant vertical velocity is called the terminal velocity, the parameter determining the severity of the fall. At first, the object keeps accelerating, thus increasing its velocity, until it reaches the terminal velocity. In other words, the object starts with the highest acceleration (gravitational acceleration, g) which drops down to zero, while the velocity increases from zero to the terminal velocity. So, if the squirrel’s terminal velocity is relatively small, the altitude it falls from will not affect its survivability from the fall.

II. Literature review and older studies

Gliding has evolved among recent mammals at least six different times [2] [3]. Among these gliding mammals, flying squirrels are probably the most aerodynamically sophisticated. They are able to modify the shape of their gliding membranes in flight, and small flying squirrels, in particular, are extremely agile and maneuverable [4] [5] [6] [7]. Wingtips of flying squirrels are held at an upward angle to the rest of the wing and form airfoils. It has been proposed that they serve the function of reducing induced drag, similar to the winglets of a modern aircraft [7] . That gliding method, along with the other techniques mentioned, helps the squirrel obtain very low terminal velocities and survive high falls.

i. Biology and anatomy of the squirrel

Before discussing the shortcomings of previous papers, in section 3.3, we must first get acquainted with the anatomy of squirrels to directly compare it when it changes shape during the glide to other objects, the plane sheet.

All flying squirrels glide with their forelimbs and hind limbs extended, between which stretches a gliding membrane, the plagiopatagium [12] [7], the "X" shape. A smaller gliding membrane, the propatagium, is supported between the cheek and the wrist in front of the forelimb. Another portion, the uropatagium, extends between the hind limbs and the tail [7]. The plagiopatagium is supported at the wrist by a finger-like projection, the styliform cartilage, which is bent back into a curve by the tibiocarpalis muscle that runs along the lateral edge of the gliding membrane [12] . This curved cartilage forms the wingtip, which, in flight, is usually held at an upward angle to the plane of the rest of the gliding membrane and resembles the winglets of modern aircraft [7]. Flying squirrels maneuver with great efficiency in the air, making 90 degree turns around obstacles, if needed [9] . Just before reaching a tree/ground, they raise their flattened tails, which abruptly changes their trajectory upwards, and point all of their limbs forward to create a parachute effect with the membrane to reduce the shock of landing [11]. The limbs absorb the remainder of the impact. When the squirrel is not gliding, the muscles and cartilages contract to return to the normal position. The cartilages get folded back and against the forearm, and the plagiopatagium gets folded against the body [7] . During the flight, the squirrel makes modifications to its aspect ratio by slightly extending or retracting its limbs. These modifications do not significantly change its cross-sectional area, but they affect the aspect ratio (during the flight). Alongside the modifications, the squirrel adjusts its tail to stabilize itself and counter any turbulence during the flight.

ii. Drag force exerted by the fluid

Factors affecting drag force

As discussed in section 2, drag force depends on many factors including drag coefficient, area, speed of the flow, etc., given by the formula:

where \(F_D\) is the drag force, \(\rho\) is the density of the fluid, \(v\) is the speed of the object relative to the fluid, \(C_D\) is the drag coefficient, and \(A\) is the cross-sectional area.

\(F_D\) also depends on the Reynolds number, \(Re\), which is given by the formula:

where \(v\) is the flow speed, \(L\) is the characteristic linear dimension, and \(μ\) is the dynamic viscosity of the fluid.

Drag force is proportional to the velocity for laminar flow and to the squared velocity for turbulent flow. Turbulent flow is considered in our paper. More importantly, drag forces always decrease fluid velocity relative to the solid object in the fluid's path [13]. Laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing [14]. At low velocities, the fluid tends to flow without lateral mixing, and adjacent layers slide past one another without mixing. There are no cross-currents perpendicular to the direction of flow, nor eddies or swirls of fluids [15]. In contrast to laminar flow, turbulence, or turbulent flow, is fluid motion characterized by chaotic changes in pressure and flow velocity [16]. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason, turbulence is commonly realized in low viscosity fluids. As follows from equation (1), drag force depends on \(C_D\), a dimensionless quantity. A lower CD indicates that the object will have less aerodynamic or hydrodynamic drag. \(C_D\) is always associated with a particular surface area [17] . CD can be also defined as a function of Re for geometrically similar bodies [18]. The Reynolds number (Equation 2) helps predict flow patterns in fluid flow situations. At a low \(Re\), flows tend to be laminar flow, while at a high \(Re\), flows tend to be turbulent. Laminar conditions apply up to \(Re = 10\), fully turbulent from \(Re = 2000\) [19]. For an object with well-defined fixed separation points, like a circular disk with its plane normal to the flow direction, the drag coefficient is constant for \(Re > 3,500\) [16] , meaning that it is considered to be in turbulent conditions. \(Re\) is a dimensionless quantity. It has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. It is used to predict the transition from laminar to turbulent flow and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full-size version. The predictions of the onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behavior on a larger scale, such as in local or global air or water movement, and thereby the associated meteorological and climatological effects.

Calculating the terminal velocity

We calculate the terminal velocity of the squirrel by equating the drag force that air exerts on the squirrel’s weight*: $$F_D = m g = \frac{1}{2} \rho v^2 C_D A$$

*\(Weight = m \times g\)

where \(m\) is mass, and \(g\) is gravitational acceleration.

Or given that \(£ = \frac{m}{A} \to m = £ \times A\) $$£ \times A \times g = \frac{1}{2} \rho v^2 C_D A$$ $$£ \times g = \frac{1}{2} \rho v^2 C_D$$

from the above we get:

where \(v\) is the terminal velocity that we want to calculate.

Prior studies

Many papers have already discussed the factors mentioned, including Alva Merle Jones’s paper [18] that discussed the C_D values for flat plates, spheres, and cylinders moving at low Re in a viscous fluid. He also discussed and illustrated the relation between drag force and velocity using different viscosities of oil (heavy and light). In addition, he discussed the relation between drag force and Re. Another paper by Berge Djebedjian, Jean Pierre Renaudeaux, et al. [20] discusses a numerical study of the drag coefficient of an infinite circular cylinder at low and moderate Re using the FLUENT code (we could have used the FLUENT code for the purpose of this paper but did not because of the high license cost). It is known that these numerical solutions depend on numerous factors including the discretization scheme, mesh sizing, boundary conditions, stability and convergence criteria that are used to obtain the solutions. That relation was confirmed through the discussion in the paper. The paper concluded that in the treated cases of a constant ratio of B/D for steady laminar flow, it is observed that the effect of the upstream distance is more pronounced than the effect of the downstream distance and the pressure drag coefficient is magnified as the numerical blockage ratio D/B increases, where B is the lateral boundary width, and the D is the diameter.

iii. Open questions left and gaps in older papers

As noticed in section 3.2, many papers focused on the factors affecting the drag force, but very few mentioned, or briefly mentioned, the effect of the aspect ratio of the body on the drag force, drag coefficient, stability, and terminal velocity. That is what will be discussed and experimented on in this paper. Aspect ratio is normally taken into consideration when crafting airplanes and aircraft. Taking into consideration the induced drag equation, there are several ways to reduce the induced drag. Wings with a high aspect ratio have lower induced drag than wings with a low aspect ratio for the same wing area. So, wings with a long span and a short chord have lower induced drag than wings with a short span and a long chord [16].

iv. Idealized model of a flying squirrel

The shape of the flying squirrel in its flight is presented in Figure 1. We simplify and idealize this shape by a rectangle of corresponding area to the total cross-sectional area of the squirrel. We thus neglect the relatively small deviations of the idealized shape, i.e., we neglect the head and tail (but take into consideration the tail’s length), which are significant for navigation, but likely not for producing the drag.

III. Thesis and Goal

Squirrels are able to survive any fall from any altitude by manipulating their drag force (F_D) and terminal velocity. They can do so by changing their shape. We believe that the main factor through which squirrels control the drag force and terminal velocity is the aspect ratio of their body shape. We will conduct experiments to test this hypothesis with our idealized models. Furthermore, we will be noticing the stability of the model as we propose that aspect ratio influences the stability of the squirrel mid-flight.

IV. Methodology

Hydrogels, polymeric nanoparticles, and carbon nanotubes all have characteristics that may be utilized to determine which nano-drug delivery method is optimal based on these properties. Sustained-release, administration methods, mechanical strength, customizability, retention, toxicity, biocompatibility, drug-carrying capabilities, and production cost are all factors to consider.

In this paper, we identified and calculated the effect of a squirrel’s aspect ratio on the terminal velocity and stability. Experiments were done on idealized squirrel paper models of the same cross-sectional area but varying aspect ratio. The dependent factor was the time and the stability of the model, and the independent factor was the aspect ratio of the body. The results were visualized in two graphs, Graph 1, and Graph 2. Five paper models for the squirrel were made, each with a different aspect ratio. Every model was tested ten times. The paper models were dropped from a drop spot h = 1.5 meters above the ground. A spot directly under the drop spot (but on the ground) was marked, and the divergence from it was noticed. The time for the fall, T, was recorded with the I-phone’s stopwatch. For each model and each trial, we calculated the terminal velocity by: $$v = \frac{h}{T}$$ We identified and noticed the stability by observing the divergence from the spot directly under the drop spot. Furthermore, we noticed the mid-fall behavior of the model and observed if there were any flips or swings. Below we list all the results in the table. Each squirrel model had the same mass (thus same weight). This was guaranteed by choosing the same paper type for each model and the same cross-sectional area. The average squirrel’s dimensions were taken into consideration when making the paper models. Southern flying squirrels have a total length (including tail) of 21–26 cm (8.3–10.2 in). The tail can be 8–12 cm (3.1–4.7 in) [21] . The average of the measurements was calculated to obtain the desired “average squirrel’s dimensions” measurements. So we ended up with 23.5 cm for their length. While gliding, the squirrel makes a square-like shape with its skin, and thus its width equals the total length minus the tail’s length (13 – 24 cm). The average of those is 18.5 cm for their width. While making the models with different aspect ratios, we got the average of those measurements. For example, in the 1:1 ratio model, the measurements were 20:20 cm (approximately the average of 23.5 and 18.5). Paper sheets that the models were made from all had the same £. This is to measure the results of the aspect ratio precisely without the interference of other factors. \(£ = 80 g/m^2\). Each model had the same area, but a different aspect ratio. The models were tested in a neutral environment of a room flow that was approximately equal in all trials. To ensure the same airflow and to minimize any changes, the room was sealed: the windows and door of the room were closed, and thus any external sources of airflow were eliminated.

Materials and Equipment

1. Paper sheets with \(£ = 80 g/m^2\), which were transformed into the squirrel models.
2. Squirrel paper models of every aspect ratio listed below in Table 1.
3. Mobile phone’s stopwatch, which was used to measure the time the models took from the drop spot to the ground.
4. Fixed bar that the models were all dropped from to ensure travelling the same vertical distance to the floor.

Measurements and accuracy

The dimensions of the paper models will be measured in cm with error ± 0.1 cm. Other variables were obtained from previous studies and calculations, such as air density, \(C_D\), etc.

V. Results

The collected experimental results are collected in Table 1, Graph 1 and Graph 2.

Let us denote the aspect ratio for this calculation (plotting the results) by the Greek letter \(η\)

$$η = \frac{L}{W}$$

• \(C_D = 0.005\) [22] .
• \(\rho = 1,225 g/m^3\) [23] .
• \(£ = 80 g/m^2\), which was obtained from the factory of the manufacturer of the paper.
• \(g = 9.8 m/s^2\)
• \(A = 400 cm^2\)

Standard deviation

• Drop time standard deviation: 0.1364
• Velocity standard deviation: 0.0841

We can calculate the theoretical terminal velocity from equation (3) :

$$v = \sqrt{\frac{2 £ g}{ρ C_D}} = \sqrt{\frac{2 \times 80 \times 9.8}{1225 \times 0.005}} = 16.0 m/s$$

VI. Discussion

The L/W = 1:1 model showed extreme stability during the first 1 meter and was often disoriented and became unbalanced during the last 0.5 meter. However, it had minor divergence from the drop spot. It landed very close under it. The L/W = 2:1 model showed extreme unbalance; the model made a 360º flip in 8 of the trials. The most divergence in the results occurred in this result; the least time and the maximum were 1.03, 1.67 respectively. However, it maintained a small divergence from the drop spot even after the 360º flips. The L/W = 3:1 model showed inconsistency in the result because, similar to the 1:2 model, it made a 360º flip in 5 trials, and in 2 trials, it made 2*360º flips. Moreover, it showed the greatest divergence from the drop spot: it moved the farthest from it. The L/W = 3:2 model, similar to the 1:1 model, showed extreme stability. It did not do any flips during all the trials. It had a slight divergence from the drop spot as well. The L/W = 4:3 showed the best results of all 5 models. It had the lowest terminal velocity, the greatest stability and the least divergence from the drop spot; it landed almost exactly under it. Considering the terminal velocity, the 4:3 model showed the best results as well. It had the lowest terminal velocity. A squirrel with such an aspect ratio would survive high falls with the smallest impact on its body. The L/W = 4:3 model showed the best results overall. It was the closest ratio to the original squirrel’s ratio, 23.5:18.5 =1.27 (4:3 =1.33). This proves that the squirrel, by its nature, is well adapted to maintain low terminal velocity and high stability during its flight, which enables it to survive in its environment and from high falls. In addition, the remarkable closeness of the aspect ratio of our best idealized model to that of a live squirrel demonstrates that the aspect ratio is indeed, as proposed earlier, the main factor for controlling the squirrel’s flight, terminal velocity and stability.

VII. Conclusion

Squirrels modify their aspect ratio during flights by extending and contracting their limbs. We proposed that these modifications affect the drag force, and thus affect the terminal velocity. Furthermore, these modifications affected the stability of the squirrel during flights. In the experiments, we tested five idealized squirrel paper models of different aspect ratios to measure the difference in the terminal velocity. We did so by measuring the time it took each paper model to reach the ground from a 1.5-meter drop spot, and the average of the times was calculated. In addition, the stability of the models was observed in all trials. This study established that the aspect ratio does, indeed, affect the terminal velocity and the drag force. We also found out that aspect ratio also affects the stability of the flight. The L/W = 4:3 model showed the best results. Its aspect ratio was the closest to the original squirrel’s ratio. This proves that the squirrel, by its nature, is well adapted to maintain low terminal velocity and high stability during its flight, which enables it to survive in its environment and from high falls. In addition, the remarkable closeness of the aspect ratio of our best idealized model to that of a live squirrel demonstrates that the aspect ratio is indeed the main factor for controlling the squirrel’s flight. The results obtained will be the groundwork for future research in the field of aircraft and airplane manufacturing. Furthermore, while this study focused on simple bodies and plane sheets proved what has been proposed, further research should focus more on complicated and sophisticated bodies that better approximate the shape of the flying squirrels and the effects of aspect ratios on them; further research should also focus on the wings of aircraft and on the shape of aircraft’s bodies.

I. Introduction

Transistors are devices made from semiconductors generally consisting of at least three terminals that amplify or switch electronic signals that were the main foundation of the electronic device. Since the mid of 20^th century, significant progress has been made in producing more effective transistors since the mid-twentieth century. John Atalla and Dawon Kahng invented the metal-oxide-semiconductor field-effect transistor (MOSFET) in 1960 to overcome surface states that blocked electric fields from the semiconductor material [1]. The effects of temperature, gate capacitance, tunnelling resistance, and source-drain voltage on current-gate voltage are precise. The simulation will offer reliable modelling for metallic single- islands SETS by monte claro way or Simulation of Nanostructures Method (SIMON). These two tunnel junctions form an isolated conducting electrode called the island, but first, we need to know what a transistor is, how it works, its development in the 20^th century, and what it was before SET.

II. The early discovery of the transistor

III. Moore's law

After that, factories have been racing to reduce the size of transistors as reduced transistor size increases the number of devices per unit area, increasing the number of operations per second. Therefore, the smallest transistor has led to increased speed, increased functional complexity, and reduced power consumption. According to Moore's Law, the number of transistors that can be put into an integrated circuit doubles every two years.

IV. SET tunnels and gate voltage

V. Conditions and challenges to prevent random tunnelling

The challenges arise because a single-electron transistor must meet certain conditions to function without random tunnelling. It must operate at a very low temperature or with very low capacitance. It is important to notice that the dimensions and structural design of the device influence the effective capacitance and resistance values and thus the device's room temperature functionality. As a result, the current emphasis is on scaling and designing the nano-islands. Even though room temperature operational single-electron transistors have been successfully fabricated, more progress toward implementing single-electron transistors that behave like conventional MOSFETs remains to be made. Researchers have turned to models and simulations to understand the behavior of single-electron transistors better. SPICE macro-modelling, Monte Carlo, and Master Equation are the three main simulation methods [6]. The character of a single-electron transistor can be modelled in SPICE macro-modelling in the same way that a MOSFET is modelled in SPICE. Although this method is significantly faster than the Monte Carlo method or the Master Equation, it does not account for the interaction of adjacent transistor devices or the Coulomb blockade effect. In addition, the master equation describes the charge transport process in single-electron circuits. Still, it becomes more challenging to solve as it is solved due to an infinite number of possible states by matrix exponential. Among the previously mentioned modelling techniques, the Monte Carlo approach was chosen for this project because it can demonstrate The Coulomb blockade effect and is more efficient than the Master Equation in computing transport configurations. This review can guide device design and pave the way for future studies of single-electron devices by examining single-electron transistor functionality.

VI. Effects in the SET

i. Tunneling effect:

ii. Coulomb Blockade Effect:

iii. Kondo effect:

VII. Methods of electron simulation

i. Method of Monte Claro

The Monte-Carlo method, macro modelling, and analytical modelling are three approaches that have been used for SET modelling. Although it is recognized that the Monte-Carlo method produces the most accurate results in SET characteristics, it is time-consuming and not suitable for mixed circuit applications. On the other hand, both macro-models and analytical models require less computation time. While the analytical Model provides direct insight into the tunneling probability of single-electron transistors, the macro Model is compact and user-friendly for circuit designers unfamiliar with single-electron transistor device physics or quantum physics [10].

ii. Simulation of Nanostructures Method (SIMON)

SIMON and other single-electron circuit simulators have procedures for calculating the charge states of all the Coulomb islands at once to account for the interaction between neighbouring Coulomb islands. These procedures are typically based on the Monte Carlo technique and necessitate a significant amount of computation time because the Monte Carlo method necessitates the calculation of average charge states in each step. Tunnel junctions, capacitors, constant voltage sources, time-dependent piece-wise linear circuits, and voltage-controlled voltage sources can be connected arbitrarily [11] . SIMON also includes two simulation models: transient and quasi-stationary modes. SIMON made the following two assumptions. The first important assumption is that voltage sources have no internal resistance. As a result, capacitance charging and discharging occur instantaneously. The simulator's second assumption is that electrons are treated as point charges that hop from island to island via tunnel junctions.

VIII. Orthodox theory

The orthodox theory predicts the IV characteristics of metallic SET devices. It is based on a semi-classical approach that includes the following assumptions: Structure and Model of the SET Device depict the device structure used in this project. The structure comprises a metal source electrode, a metal island, a metal drain electrode, and a metal gate electrode. There is also a tunnel junction between the metal source electrode and the metal island and a tunnel junction between the metal island and the metal drain electrode. The source electrode is powered by a voltage source V, while the gate electrode is powered by a different voltage source Vg. The drain electrode is connected to a grounded conductor. The SET device's physical Model is the same.

IX. Monte Carlo Predictions on Quantum Computers

The Monte Carlo method begins with all possible tunnelling events. The tunnelling events' probabilities are computed. The probabilities of the events are weighted, and one possible event is chosen at random. This random event is based on many processes such as subcircuit processing and Gaussian random number generator that is illustrated as follows:

X. Subcircuit processing

Subcircuits are reusable circuit element blocks. They are specified just once and can be used again throughout the netlist, including from different subcircuits. Because of their reusability, subcircuits require special consideration [12] . A subcircuit definition's randomization is insufficient because the subcircuit may be used multiple times within the netlist, and each instance of a subcircuit must be uniquely randomized. Because of this requirement and the fact that subcircuits may refer to one another, a recursive parser is required to interpret the netlist properly. A parser of this type is used allowing for the recursive generation of unique subcircuit instances. Every reference to a subcircuit, whether in the netlist's body or within another netlist, is replaced by a reference to a uniquely randomized copy of the subcircuit.

XI. Gaussian random number generator

In the descriptions of various Gaussian random number generators (GRNG) algorithms, we assume the existence of a uniform random number generator (URNG) capable of producing random numbers with uniform distributions over a continuous range (0, 1). The range excludes 0 and 1 because each is potentially an invalid input for a GRNG; for example, the Box-Muller method requires a non-zero URNG input, and CDF inversion requires a URNG input that is strictly less than 1. When an algorithm uses multiple samples from a uniform random number generator, the different samples are identified with subscripts. All random numbers within the loop body are generated from scratch for each loop iteration in algorithms with loops. In an algorithm, for example, U₁ and U₂ represent two independent uniform samples [13] . The probability density function (PDF) of a Gaussian distribution with mean zero and standard deviation one, also known as a "standard normal distribution" is:

$$\phi (x) = \frac{1}{\sqrt{2 \pi}} e^{-x^2/2}$$

Equation 1. probability density function (PDF) of a Gaussian distribution with mean zero

A plot of $\phi (x)$ versus $x$ gives the familiar bell-curve shape but does not directly indicate the probability of occurrence of any particular range of values of $x$. Integrating the PDF from $-\infty$ to $x$ gives the cumulative distribution function (CDF):

$$\phi(x) = \int_{-\infty}^{x} \phi(x) dx = \frac{1}{2} [1 + erf(\frac{x}{\sqrt{2}})]$$

Equation 2. Getting the cumulative distribution function (CDF) by integrating the PDF from −∞ to x.

The CDF ($x$) expresses the possibility that a random sample from a Gaussian distribution will have a value less than $x$. The CDF can be used to determine the probability of values occurring within a given range, such as the probability of a number between a and b occurring. (where a < b) is $\phi$ (b) $-$ $\phi$ (a). There is no closed-form solution for $\phi$ or for the related function erf, so it must be calculated numerically or using some form of approximation

XII. The CDF Inversion Method

CDF inversion works by taking a random number $\alpha$ from $U$ (0, 1) and generating a Gaussian random number $x$ through the inversion $x = \phi ^{-1} (\alpha)$. Just as $\phi$ associates Gaussian numbers with a probability value between zero and one, $\phi ^{-1}$ maps values between zero and one to Gaussian numbers [14] . While this is conceptually simple and exact if $\phi ^{-1}$ is calculated correctly, the lack of a closed form solution for (-1) for the Gaussian distribution necessitates the use of approximations, which affects the quality of the resulting random numbers. Because increased accuracy necessitates increased complexity, the majority of research in this area has focused on improving this trade-off. Numerical integration provides arbitrarily high precision at a computational cost that makes it unsuitable for random number generation, particularly in the Gaussian tail regions. As a result, polynomial approximations are used in the majority of Gaussian CDF inversion methods.

XIII. Transformation Methods

i. Box-Muller Transform

One of the earliest exact transformation methods is the Box-Muller transform. It generates two Gaussian random numbers from two uniform numbers. It makes use of the fact that the 2D distribution of two independent zero-mean Gaussian random numbers is radially symmetric if their variances are the same. This is easily demonstrated by multiplying the two 1D distributionse $-x^2 \: e^{-y^2} = e^{-(x^2 + y^2)} = e^{-r^2}$. The Box-Muller algorithm can be thought of as a method in which the output Gaussian numbers represent two-dimensional coordinates. The magnitude of the corresponding vector is obtained by transforming a uniform random number; the random phase is obtained by scaling a second uniform random number by $2 \pi$ [15]. Following that, Gaussian numbers are generated by projecting them onto the coordinate axes. This method is carried out using pseudo-code provided by Algorithm 1. Because the algorithm generates two random numbers each time it is run, it is common for a generation function to return the first value to the user while caching the second value for later use in the next function call. The polar variant of the Box Muller Transformation is used to create the random numbers. Given two independent and uniformly distributed variables $X_1$ and $X_2$, Gaussian distributions $Z_1$ and $Z_2$ are created:

$$y = x_1^2 + x_2^2\: , \, y\forall (0,1]$$

$$z_1 = x_1 \cdot \sqrt{\frac{-2\ln(y)}{y}}$$

$$z_2 = x_2 \cdot \sqrt{\frac{-2\ln(y)}{y}}$$

Equation 3. II. Gaussian random number equation

ii. Central Limit Theorem (Sum-of-uniforms)

Convolving the constituent PDFs yields the PDF describing the sum of multiple uniform random numbers. Thus, according to the central limit theorem, the PDF of the sum of K uniform random numbers $V/2$ each over the range $(-0.5,0.5)$ will approximate a Gaussian with zero mean and standard deviation $\sqrt{\frac{K}{12}}$,, with larger values of $K$ providing better approximations [16]. The main disadvantage of this approach is that as $K$ increases, the convergence to the Gaussian PDF becomes slower. Realizing that the sum is bounded $K/2$ and $K/2$, and that the PDF of the sum is composed of segments that are polynomials with degrees limited to $K-1$ provides some intuition. As a result, the Gaussian approximation in the tails is particularly poor.

iii. Piecewise Linear Approximation using Triangular Distributions

Due to the fact that the component distributions are triangles, only addition and multiplication are required. Outputs are generated by randomly selecting one of the triangle distributions and then generating a random number from within that distribution. Walker's alias method is used to select triangles from a discrete distribution using one uniform input; triangle distributions are then generated using the sum of two more appropriately scaled uniform inputs [17]. This method has the disadvantage of requiring three random numbers per output sample, which makes it quite computationally expensive to implement in software. However, because uniform random numbers are relatively cheap to generate in hardware, while multiplications and other operations are more expensive, this method is more appealing. This method can provide an efficient Gaussian random number generator in hardware by using a large number of triangles and the central limit theorem to combine multiple random numbers.

XIV. Conclusion

The presented tool makes performing Monte Carlo analysis on analogue circuits easier by automating the generation of many randomized netlists, their simulation, and statistics extraction from the simulation data collection. Various mismatch models are used to randomize circuit components. Based on user-specified parameters, linear circuit components are varied using a Gaussian distribution for component value. The most important factors for MOSFETs are threshold voltage and current factor mismatches. This transistor is randomized by connecting an ideal voltage source in series with the gate to represent threshold voltage mismatches and a current-controlled current source in parallel with the drain and source to represent current factor mismatches. The simulation results of four different circuits are presented, along with a discussion of the benefits and drawbacks of the techniques presented.

I. Introduction

A. One-dimensional photonic crystals

A. Anti-glare lenses

One-dimensional photonic crystals can be used as a coating on low reflection or anti-glare lenses [6]. Low-reflection glasses coated with one-dimensional photonic crystals can reflect wavelengths of 400-500 nanometers (blue wavelengths). Additionally, these lenses can effectively increase the absorption of other wavelengths of visible light. Due to the shortness of the blue wavelength, it comes into collision with dust and air molecules found in the air, which causes it to scatter [7]. As a result of this scattering, the contrast lowers. These low-reflection coating are very useful for people that who have undergone retina surgery [8]. Moreover, they can reflect light that is coming from cars approaching the driver [9] .

II. Design And Calculations

A. Refractive indexes of the alternating layers

Consequently, in this paper the \(\ce{SiO2}\) refractive index value will be \(n_1= 1.4661\) from (11). Moreover, the \(\ce{TiO2}\) value will be \(n_2 = 2.4358\) from (14). The variance of \(\ce{TiO2}\) and \(\ce{SiO2}\) are represented in the following graphs.

B. Calculating each layer’s thickness

C. Transfer matrix method.

The transmission will be calculated and plotted using the transfer matrix method [16]. Each “stage” that light passes through is represented by a matrix [17] , where the general formula for the matrix that represents the light’s propagation through a certain layer of glass of refractive index \(n_x\) and thickness \(d_x\) is [18]: The boundary matrix that light propagates from medium of refractive index $n_x$ to a medium of refractive index $n_y$ has the following general formula [19]: Furthermore, to calculate the M matrix which models the transmission of the entire one-dimensional photonic crystal, the previously calculated matrices are multiplied as the following [20]: Where N is the number of layers of the one-dimensional photonic crystal designed, which is the only unspecified variable in this matrix. As a result of calculating the M matrix, the transmission value for each wavelength can be obtained. If the M matrix consists of the following components: Then, the transmission equals the absolute of the inverse of \(M_{22}\), as \(M_{22}\) is a complex number [21]. By using the thicknesses and refractive indexes stated previously and doing the necessary calculations, a photonic band gap will exist at 450 nanometers, which is the targeted wavelength to be removed. However, the number of layers of the one-dimensional photonic crystal is not specified yet, which will be chosen due to certain limiting factors. The previously mentioned equations and matrices were coded using python, and by using the library matplotlib.pyplot, the transmission was plotted for each wavelength between 300 and 800 nanometers.

III. Photonic band gap width and transmission behavior

A. The number of layers: N

Choosing the wavelength to be reflected = 450 nanometers, several tests were conducted to observe the behavior of the width of the photonic band gap by increasing and decreasing the number of layers: N. The transmission-wavelength relation was plotted for each wavelength from 300 to 800 nanometers, and the photonic band gap width was measured at 50% transmission for each number of layers [22]. It was observed that by increasing the number of layers, the transmission of the wavelengths in the range of the band gap width (\(T_{g}\)) decreased: This is observation will be very useful in designing and engineering photonic band gaps. If the desired band gap is from 400 to 500 nanometers of wavelength, the magnitude of transmission in those wavelengths can be minimized by increasing the number of layers of the one-dimensional photonic crystal. Therefore, the number of layers is a significant factor in engineering the optimum design for a one-dimensional photonic crystal.

B. The difference between the refractive indexes: \(n_{1}, n_{2}\)

Moreover, another important factor affecting the band gap width is the refractive indexes of the alternating layers in the one-dimensional photonic crystals [23]. To observe the behavior of the band gap width by changing the refractive indexes of the materials used, several tests were conducted by changing the refractive index of the second material (\(n_2\)) while keeping the refractive index of the first material (\(n_1=1.4661\)), which is \(\ce{SiO2}\), constant. The transmission of each wavelength within the range 300 to 800 nanometers was plotted again at \(N = 5\), and the photonic band gap width was measured at 50% transmission for each number of layers. It was observed that by decreasing the difference between the refractive indexes (\(D_{n_1,n_2}\)) the total amount of transmission within the gap increased: This a very important factor in engineering the one-dimensional photonic crystals, as increasing the difference between the refractive indexes of the layers increases the reflection of the wavelengths in the band gap range.

IV. Fabrication and tolerance

A. The high thickness (N) of the one-dimensional photonic crystal

As mentioned previously, increasing the number of layers N significantly increases the reflection in the chosen range of wavelengths. However, having N = 500 would mean that there are 1000 layers of alternating silicon dioxide and titanium dioxide having a total thickness of about 124 micrometers. This can provide problems in the fabrication of the one-dimensional photonic crystal, as it is time consuming [24]. Moreover, a design that comprises the least total thickness while having the highest reflection in the chosen range of wavelengths should be pursued.

B. The tolerance in refractive indexes

C. Thickness approximation

V. Optimization of thickness using genetic algorithm

A. Approach and basic optimization

This paper discusses an approach to creating a one-dimensional photonic crystal with minimum thickness and number of layers. Therefore, several methods of decreasing the thickness of the one-dimensional crystal while maintaining an increase in reflectance between 400 and 500 nanometers were sought after and investigated. The number of layers of the one-dimensional crystal is N = 5, 5 layers of silicon dioxide and 5 layers of titanium dioxide. Moreover, the thickness of each layer will be variable. The aim of this paper is to find which thickness value of each of the ten layers produces the best results. The best results are determined based on the transmission within the band gap, from 400 to 500 nanometers, and the transmission outside the band gap range within the visible spectrum. The transmission within the band gap should be minimized, while the transmission outside the band gap range should be maximized, effectively reflecting blue wavelengths while passing the other colors' wavelengths. If only the thickness of each alternating layer was variable, creating only two variables with the need to be optimized, it could be optimized using a for loop. However, to produce more efficient results, the thickness of the 10 layers of alternating materials will be optimized. However, 10 variables cannot be optimized using for loops. Consequently, genetic algorithm optimization using Matlab is used. A code is written with the thickness of each layer being specified as the genetic algorithm optimization function input, meaning that the initial population contains various values of thickness. Moreover, the bounds for the values of the initial population vary between a lower bound of 10 nanometers and an upper bound of 500 nanometers. After the thickness values are inputted, a for loop runs from 370 to 700 nanometers, calculating the transmission value for each wavelength within the visible spectrum [25] using the transfer matrix method. Furthermore, an if statement is written within the for loop, stating that if the wavelength that the transmission value is being calculated for is between 400 and 500 nanometers, append the value of the transmission for that wavelength within the array \(T_g\), and if the wavelength is out of that range then append its transmission value within the array \(T_d\). After the for loop completes calculating the transmission values for all wavelengths within the visible spectrum, the mean of both the arrays \(T_g\) and \(T_d\) is calculated. Moreover, a new variable named \(T_x\) is specified as having the value of the ratio: \(\frac{T_g}{T_d} \). Another variable called \(T_{k}\) is specified to equal the total thickness of the one-dimensional photonic crystal. Moreover, the value of the function f which is going to be optimized equals: This means that the best individual result of the genetic algorithm code will have the value closest to zero. Furthermore, as shown in the equation, the total thickness of the one-dimensional photonic crystal, as a variable in the fitness function, was given a weight of 10%, with the rest of the weight being given to the transmission behaviour within the band gap.

B. The effect of selection method on the best individual

The method of selection will greatly affect the result of the best individual. Selection in evolution is the natural process by which a parent is chosen for the next generation [26]. There are several methods of selection available in Matlab [27] . Three options' behavior will be studied. These three methods are Roulette selection, Stochastic selection, and Remainder selection. The Roulette method of selection is just like a wheel that is separated into several segments [28]. These segments' area is proportional to the expectation of the respective parent [29]. Moreover, a random number is generated to select a segment of the wheel with the possibility being also directly proportional to the area of the segment of the Roulette wheel [30]. The Stochastic method of selection works by creating a line. The line is separated into several segments, with each segment corresponding to a parent [31]. The length of each segment is directly proportional to the expectation of that respective parent. Along that line segment, the algorithm moves one step for each parent with the steps being equal [32]. When the algorithm lands on a parent, it allots that parent. Moreover, a random number is generated less than the step size, and its value is assigned to be the first step [33]. The Remainder selection method assigns the parent according to the integer value of their scaled value [34]. Then the Roulette selection method is used on the remaining fractional part. These three selection methods were tested using the genetic algorithm to see which would produce the best results. They were tested with the mutation method being Uniform, the crossover being Intermediate, and the initial population to be 250 individuals. The genetic algorithm optimization was tested throughout 1000 generations to produce the best possible individual. Since the function that is being optimized has a value stated in equation (10), then the ideal value of the best individual is zero. Moreover, as shown in figures 14, 15, and 16, the best individual resulting after 1000 generations of each selection method had a very good fitness value. Furthermore, it was observed that the Roulette and Stochastic methods had very similar results with the resulting best individual having a fitness value of 0.171762 and 0.171761 respectively. The thickness value of each layer of the best individual in both the Roulette and the Stochastic selection methods is nearly the same. The Remainder method of selection had a noticeably worse fitness value than both the Roulette and the Stochastic selection methods. It was also observed that the three methods of selection had a periodic thickness, with only two layers being higher or lower than that periodic value. As shown in the figures representing the plots of the best individual of each selection method, the Stochastic method of selection produced the best results with the Roulette method of selection ranking second by only a very slight margin.

C. The effect of the mutation method on the best individual

Mutation was yet another crucial factor in the genetic optimization of the thickness of the one-dimensional photonic crystal. A mutation is a mechanism by which the algorithm makes small changes in the individuals to promote diversity [35] . In real-life evolution, genetic diversity is extremely crucial for evolution to take place effectively, as the best individual has better chances to appear in a genetically diverse environment [36]. Three methods of mutation were tested and the differences in their results emphasized which method was most effective and which was least effective. The three methods were the Uniform, Gaussian, and Adaptive Feasible mutation methods. The Uniform mutation with a rate of 0.5 produced extremely suitable results with three different selection methods as shown in figures (14, 15, and 16). The best combination was illustrated in figure (15), which implemented the Stochastic selection method as well as Uniform mutation and the Intermediate crossover method. Consequently, the Stochastic selection method will be the selection method used while testing the three methods of mutation mentioned previously. The Uniform mutation method consists of two steps. During the first step, the algorithm takes a fraction of the numerical value of the individual, and the probability of the individual undergoing mutation is the same as the rate of the mutation specified, which in this case is 0.5 [37]. In the second step, the algorithm replaces this fraction with a randomly generated number. The results of the Uniform mutation method combined with Stochastic selection and Intermediate crossover are shown in figure (15) and its plot is shown in figure (18). The Gaussian mutation method works by adding a random number to the individuals [38]. From a Gaussian distribution centered at zero, that number is taken. Moreover, during the testing of the Gaussian mutation method, the best individual had values that were out of the bounds specified, from 10 to 500 nanometers. Some of its thickness values were negative, which is impossible. Consequently, it was neglected while testing the rest of the mutation methods as well as the crossover methods. Furthermore, the Adaptive Feasible mutation method was also tested, combined with Stochastic selection and Intermediate crossover. The Adaptive Feasible mutation method makes changes to the last generation, whether it was successful or unsuccessful, that are completely random but within the chosen bounds [39]. The Adaptive Feasible method produced considerably inferior results to the Uniform mutation method, as it had the best individual with a fitness value of 0.541. After assessing the results, the Uniform mutation produced the best individual and proved to be the most efficient mutation method.

D. The effect of the crossover method on the best individual

Crossover is the third mechanism affecting the results of the optimization of the thickness of the one-dimensional photonic crystal using the genetic algorithm. To test which method of crossover produces the individual with the best results, three different crossover methods were tested with the previously determined best methods of mutation and selection, which were Uniform mutation and Stochastic selection. These three methods of the crossover were Arithmetic, Intermediate, and Scattered crossover. The Intermediate crossover method works by creating an average between the parents of the individual [40] . The weight of this average is determined by a random fraction value [41]. The child individual generally lies between the value of both parents. Furthermore, the Arithmetic crossover method is very similar to the Intermediate crossover method in terms of operation. The Arithmetic method of crossover generates a random average value of the two parents lying on the line between them and assigns it to the child individual [42]. Moreover, the Scattered method of crossover is the most different method. It works by generating a random binary vector and then extracts the genes where the vector was 1 from a parent and the genes where the vector was 0 from the other parent [43]. Figure (15) represents the results of the genetic algorithm optimization over 1000 generations using the combination of the Stochastic selection method, Uniform mutation, and Intermediate crossover. The best individual had a fitness value of 0.172 approximately. Moreover, figures (22 and 23) represent the results of the combination of Arithmetic crossover and Scattered crossover respectively with Stochastic selection and Uniform mutation. The Arithmetic crossover combination had a best individual with a fitness value of 0.170049, and the Scattered crossover had a best individual with a fitness value of 0.170594, making the Arithmetic crossover slightly better than the Scattered crossover method.

Selection	Mutation	Crossover	Best individual's fitness value
Remainder	Uniform	Intermediate	0.174912
Roulette	Uniform	Intermediate	0.171762
Stochastic	Adaptive Feasible	Intermediate	0.541335
Stochastic	Uniform	Arithmetic	0.170049
Stochastic	Uniform	Intermediate	0.171761
Stochastic	Uniform	Scattered	0.170594

VI. Conclusion

In conclusion, this paper aimed to find the best possible design for a one-dimensional photonic crystal. The photonic crystal would reflect light from 400 to 500 nanometers to be used for manufacturing anti-glare lenses. The transfer matrix method was investigated. Moreover, the effect of the number of layers and the difference between the refractive indexes of the alternating layers were calculated and documented. Fabrication errors were also considered, such as thickness estimation and refractive index deposition errors. A python code was written to simulate the results of the photonic crystals and assist in investigating their properties. Moreover, it was noticed that the best results were produced from photonic crystals with the highest relative thickness. To overcome this, global optimization of the thickness of a 5-layered one-dimensional crystal was implemented in Matlab using a genetic algorithm. Several selection, mutation, and crossover methods were investigated to find the best possible combination. It was observed that the best individual resulted from the combination of Stochastic selection, Uniform mutation, and Arithmetic crossover. The fitness value to determine which individual was the best was calculated based on two factors: thickness and transmission. The transmission factor is the ratio of the transmission within the band gap to the transmission outside the band gap, and it had a weight of 90%. Moreover, the thickness factor was calculated by dividing the total thickness of the photonic crystal by the maximum thickness, 5000 nanometers, which was determined by the stated bounds in the Matlab code. The best individual had a fitness value of 0.17 with a periodic structure, with only the first and last layer having considerably larger thickness values. In conclusion, this paper designed an effective one-dimensional photonic crystal to be used in manufacturing anti-glare lenses.

I. Introduction

From the genesis of humans, the idea of infinity has been at the centre of attention and constantly stirred the emotions of humanity more deeply than any other question. Philosophers have been attracted, fascinated, perplexed, and disturbed by the infinity in its many different guises, by the endlessness of space and time, by the thought that between any two points in space, no matter how close, there is always another, by the fact that numbers go on forever, and by the idea of the infinite perfection of god. Infinity is one of the most perplexing notions to scientists and philosophers because it is difficult to find in empirical reality; even as an idea or concept, it is beyond our consciousness and mental capabilities. At first, mathematicians tried to avoid the notion of infinity or even demonstrated hostility throughout most of the history of mathematics. However, now it is one of the most important mathematical and philosophical postulates and one of the most essential rules in the conceptual development of mathematics especially the development of calculus. Greeks in the fifth and sixth centuries B.C faced much confusion with infinity; even they called it “Aperion” [1]. Apeiron was a bad, even derogatory term. Apeiron was the primal chaos from which the world emerged. Apeiron was an arbitrarily crooked line. As a result, Aperion meant completely disorganized, indefinitely complicated, and susceptible to no limited determination. Pythagoras was one of the most prominent opponents of the concept of infinity, and anybody who tried to declare the existence of irrational numbers would be killed by his disciples, as Hippasus was. Infinity has only been a part of mainstream mathematics since Georg Cantor's (1845-1918) work, and his theories were first received with criticism since they were not supported by what we perceive in the real world around us. Prior to Cantor's time, mathematicians had wrestled for millennia with the concept of infinite, generally without success. Indeed, the ancient Greeks' preference for geometry over algebra can be linked in part to their difficulties with infinite processes. For example, the square root of two may be produced geometrically in a few steps, but defining it algebraically requires knowledge of an infinite technique. Infinity has a lot of paradoxes and contradictions that make accepting it more complicated. However, the questions are: How can human mind generate the idea of infinity, and How can we mathematically prove the existence of infinity?

II. Attempts to contest the idea of infinite

Initially, infinity was merely associated with extremely lengthy time periods, great distances, or enormous sets. Ancient Greeks were aware of the infinite issue, despite the fact that the idea of infinity is inherently exceedingly difficult for a human mind to grasp or even, as some claim, beyond human comprehension. This is confirmed by the history of disputes. There have been countless paradoxes and dilemmas regarding the concept of infinite from the beginning of humanity. Most people are familiar with Zeno of Elea (490–430 B.C.) from his extraordinarily complex paradoxes of infinity.

i. Zeno's paradoxes

Zeno’s paradox of motion is mainly four paradoxes that have been solved in 2003 which are, (Achilles and the tortoise, The dichotomy, the arrow, and the stadium). [2]

i.i. The Achilles and the tortoise

i.ii. The dichotomy

In order to reach the finish point, the Achilles should first reach the half of the way. Before reaching the end point, the Achilles should reach the half of half of the way. This process will be repeated until reaching the finish point. But when will the Achilles reach the finish point?? In a mathematical way, Define a sequence $a_{n}$ where $$a_{n}=\bigg\{\frac{1}{2^n}\bigg\}_{1}^{\infty}=\frac{1}{2},\frac{1}{4},\frac{1}{8}, \dots \qquad eqn 1$$ The sum of this infinite series is 1 where $$\sum_{n=1}^{\infty}\frac{1}{2^n}=1 \qquad eqn2$$ But, when will the value 1 will be reached, the infinity cannot be reached. There is no last member in the sequence or the series. In the regressive version of the paradox, it illustrates that an Achilles cannot began the race or even move such a single meter. The Achilles before running the first quarter, he should run the first eighth, and so on. So, when will the Achilles start running. In a mathematical way, Define a sequence $b_{n}$ where, $$b_{n}=\dots ,\frac{1}{8},\frac{1}{4},\frac{1}{2} \qquad eqn3$$ This type of infinite series has no first member. So, it is impossible that the Achilles start running.

i.iii. The arrow

ii. The painter's paradox

It is a geometry paradox that depends mainly on the limitless surface area and finite volume of Gabriel's horn that serve as the foundation for the Painter's Paradox. When we assign limited contextual meanings of area and volume to an ethereal object like Gabriel's horn [3], the paradox occurs. Define a function $f(x)=\frac{1}{x}$ by using the surface, and the volume of revolution of this function, we have the volume $$V=\pi\int_{1}^{\infty}(f(x))^2 dx \qquad eqn4$$ While the surface area $$A=2\pi \int_{1}^{\infty}f(x)\sqrt{1+(f(x))^2}dx \qquad eqn5$$ Solving to get the volume of revolution of this function $$V=\pi\int_{1}^{\infty}(f(x))^2 dx$$ $$V=\pi\int_{1}^{\infty}(\frac{1}{x^2}) dx$$ $$V=\pi \lim_{a \to \infty} \bigg[-\frac{1}{x}\bigg]_{1}^{a}$$ Where $\frac{1}{\infty}=0$ So, $$V=-\pi \lim_{a \to \infty} \bigg(-\frac{1}{a}-1\bigg)=\pi$$ However, there is a sudden result for the surface area $$A=2\pi \int_{1}^{\infty}f(x)\sqrt{1+(f(x))^2}dx$$ $$A=2\pi \int_{1}^{\infty}\frac{1}{x}\sqrt{1+\frac{1}{x^2}}dx$$ $$>\int_{1}^{\infty}\frac{1}{x}=2\pi \lim_{b \to \infty} \bigg[\ln(x) \bigg]_{1}^{b}$$ $$=2\pi \lim_{b \to \infty} (\ln(b)-\ln(1))=\infty$$ This results means that Gabriel's horn is possible to be filled without covering its surface area completely which seems to be impossible.

iii. Two concentric circles paradox

Galileo Galilei presented an intriguing answer to this problem in the early 1600s. Galileo suggested that an infinite number of infinitely small gaps may be added to transform the shorter length into the greater length. He was fully aware that such a technique results in a variety of difficulties: "These difficulties are real; and they are not the only ones. But let us remember that we are dealing with infinites and indivisibles, both of which transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness. In spite of this, men cannot refrain from discussing them, even though it must be done in a roundabout way." [4] He overcame some of his difficulties by saying that problems happen just once "when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.".[5] This last claim is backed by an example known as Galileo's paradox.

iv. Galileo's paradox

In Galileo's paradox of infinity, the set of natural numbers, $\mathbb{N}$, is compared to the set of squares, $\{n^2: n \in \mathbb{N} \}$. It seems obvious that most natural numbers are not perfect squares, so the set of perfect squares is smaller than the set of all natural numbers. On the other hand, Galileo asserts a one-to-one correspondence between these sets, and on this basis, the number of the elements of $\mathbb{N}$ is considered to be equal to the number of the elements of $\{n^2: n \in \mathbb{N} \}$. This creates a paradoxical situation. It means that we cannot treat infinities as we treat finite sets and that they cannot be compared in terms of greater-lesser. The conclusion of this paradox for Galileo was that "we can only infer that the totality of all numbers is infinite, and that the number of squares is infinite $\dots$; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally, the attributes 'equal,' 'greater,' and 'less,' are not applicable to infinite, but only to finite quantities." [6]

v. Hilbert’s Paradox of the Grand Hotel

However, what if this guest has a companion with him? We will follow the same procedure, but instead of asking the visitor in room one to go to room number 2, we will tell him to go to room number 3 and ask the guest in room number 2 to go to room number 3 by continuing this pattern we will have room one and two empty. We can expand this idea further. If 50, 100, or even 10,000,000 tourists arrive at our hotel, we can use the same approach. Notice that You could always get higher in the counting since there are an infinite number of rooms; there is no such thing as the last room. However, assume that an additional infinite number of visitors arrive and request rooms. Will we then construct an additional infinite hotel to house them? According to the aforementioned fundamental theorem of arithmetic, each person must go to their own room. No accommodation will be double booked, and no one will be without a room. There will also be many vacant rooms, for instance, the number 15 cannot be expressed as a product of twos and threes. The crucial thing is that everyone of the visitors has been assigned their own room. The beauty about Hilbert's hotel is that we can keep going with more cases. Assume now that an infinite number of planes arrived, each bringing an infinite number of buses and an infinite number of passengers. To accommodate these visitors, we may extend our earlier strategy and employ the next prime number, 5. So the passenger in seat n, bus c, and plane f will now enter the room numbered $$2^n \times 3^c \times 5^f \qquad eqn7$$ In our completely booked hotel, each guest will have their own room. The Greeks as a whole despised the idea of infinity, and Plato (428/427-348/347 BCE) and Aristotle (384-322 BCE) both shared this opinion. Aristotle outlined two perspectives on an infinite series. The actual infinite and the potential infinite stand out as two unique perceptions [7]. The infinite is treated as eternal and complete, as something that existed entirely at one point in time, according to the idea of actual infinity. The idea of actual infinity views the infinite as a continuously evolving, non-terminating process. Potential infinity is a method that approaches, but never quite achieves, an infinite end. For example, consider the number sequence. $$ 1, 2, 3, 4, \dots$$ It grows higher and higher, yet it never reaches infinity. Infinity is simply a pointer in the right direction. Actual infinity is an infinity that one truly attains; the process is complete. Let's place braces around that previously described sequence: $$ \{1, 2, 3, 4, \dots \}$$ We are representing the set of all positive numbers with this notation. There is just one item in this set. But the members of that set are infinite. Furthermore, Aristotle believes that actual infinity cannot exist even as a notion in the mind of a human. Mathematicians in the 17th century began to deal with abstract things that did not correlate to anything in nature. Mathematics was able to depict the behavior of nature more properly by employing the idea of infinitesimal numbers, or numbers that are infinitely tiny. In the late 1600s, the English mathematician Isaac Newton and the German mathematician Gottfried Wilhelm Leibniz discovered calculus which was based on the idea of infinitely tiny numbers [8]. In order to support the computation of derivatives, or slopes, Newton developed his own theory of infinitesimals. He discovered that it was helpful to look at the ratio between $dy$ and $dx$, where $dy$ is an infinitesimal change in $y$ generated by moving an infinitesimal amount $dx$ from $x$, in order to estimate the slope for a line contacting a curve at a given point $(x, y)$. Massive criticism of infinitesimals resulted in early analysis's focus on trying to establish an alternative, robust theoretical framework for the concept. After Abraham Robinson, a mathematician of German descent, developed nonstandard analysis in the 1960s, the usage of infinitesimal numbers finally found a solid foundation. It was Bernard Bolzano (1781–1841) who first advocated the use of the idea of infinity in mathematics. According to him, the universe is filled with actual infinities, hence it is absurd to leave them out of mathematics. 14 Bolzano asserts that a complete infinite may be envisioned by the human intellect. For two and a half millennia prior to George Cantor (1845–1918), infinity was the topic of muddled reasoning and flawed arguments. A mathematical theory developed by Cantor completely restored the actual infinite, answered the previous paradoxes.Certainly, Cantor was the first to realize that there are several types and sizes of infinite. Cantor himself was so taken aback by what he discovered that he wrote: "I see it, but I don’t believe it!" [9] Infinity has previously been considered a forbidden concept in mathematics. According to some, such as Johann Carl Friedrich Gauss (1777-1855), the prince of mathematics, and one of the greatest of them, the term "infinity" should only be used as a "way of speaking," [10] in which one legitimately speaks about boundaries, but not as a mathematical term. Gauss wrote to Schumacher on July 12, 1831, in a letter: "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction." [11]

III. Cantor and his work on infinity

Cantor first was successful in demonstrating the uniqueness theorem for any function. $$\sum_{n}(a_{n}\sin(nx)+b_{n}\cos(nx))\qquad eqn8$$ in the case of convergence of the trigonometric series for all values of $x$. After that, in 1871, Cantor wrote a note demonstrating that, provided the number of such exceptional points x was finite, the uniqueness theorem held true even if, for specific values of x, either the representation of the function or the convergence of the trigonometric series were finite. A year later, Cantor published his most important work on these topics—a big paper demonstrating that, provided that the distribution of the exceptional points was specific, the uniqueness theorem held true even in the case of an unlimited number of such points. He developed the theory of point sets as well as his later theory of the transfinite ordinal and cardinal numbers. In his paper from 1872, he introduced a specification that dealt with the limit points of an infinite set $P$. The Bolzano-Weierstrass theorem states that each such set must have at least one point, and that any arbitrarily small neighbourhood of such a set includes an unlimited number of points. The set of all such limit points of $P$ Cantor denoted $P'$ the first derived set of $P$. If $P'$ also contains an infinite number of points, it too must contain at least one limit point, and the set of all its limit points $P''$ is the second derived set of $P$. Continuing his consideration of derived sets, if for some finite number $v$ the $vth$ derived set $P^{(v)}$ is not an infinite set, then its derived set, the $(v+1)th$ derived set of $P$,$P^{(v+1)}$, will be empty. Then, $P^{(v+1)}= \phi$. Cantor called such sets derived ‘sets of the first species’, and for such sets of exceptional points of the first species, he was able to show that his uniqueness theorem for trigonometric series representations remained valid. As yet, he did not know what to make of derived sets of the second species, but these would soon begin to attract his attention, with remarkable and unexpected consequences. In 1880, Cantor published the second article in his series on linear point sets, which featured his transfinite numbers for the first time. It produced an infinite series of derived sets, starting with an infinite set P of the second species: $$P',P'',P''', \dots ,p^{(v)}, \dots \qquad eqn9$$ Cantor defined the intersection of all these sets as $P^{\infty}$.However, if $P^{\infty}$ was infinite, it then gave birth to the derived set $P^{\infty +1}$, which resulted in a new series of derived sets. Assuming all of the subsequent derived sets were infinite, then the following sequence of derived sets was possible: $$P',P'',P''', \dots ,P^{(v)}, \dots ,P^{\infty}, P^{\infty +1}, \dots \qquad eqn 10$$ Cantor continued to concentrate on the sets themselves rather than the "infinite symbols" He utilised symbols to designate each of the subsequent derived sets starting with $P^{\infty}$. He would soon start to recognise these symbols as transfinite ordinal numbers. Finally, Cantor understood that his "infinite symbols" might be viewed as true transfinite numbers that were mathematically equivalent to the finite natural numbers, rather than merely being indices for derived sets of the second species. According to him, "I shall define the infinite real whole numbers in the following, to which I have been led over the previous several years without recognising that they were concrete numbers with genuine significance." Cantor used two principles of generation to create his new transfinite ordinal numbers apart from the derived sets of the second species. The first concept was the expansion of the well-known series of natural numbers, which began with the repetitive addition of units and goes on to include $1, 2, 3, \dots$ It was feasible to imagine a new number, $\omega$, that conveyed the natural, regular order of the full series of natural numbers despite the fact that this sequence lacked a biggest member. This new number came after the complete series of natural numbers $v$ and was the first transfinite number. After defining, $\omega$ it was able to use the first principle of generation once more to create a new series of transfinite ordinal numbers, as seen in the following: $$\omega, \omega+1, \omega+2 , \dots, \omega+v, \dots \qquad eqn11$$ Again, since there was no largest element, it was possible to introduce another new number, $2 \omega$, coming after all the numbers in the above sequence, and in fact representing the entire sequence. Cantor explained his second principle of generation, adding new numbers whenever a given sequence was limitless. [12] By successfully applying the two principals shown above, it is possible to show a new infinite sequence of number, The most general term began $$v_{0}\omega^{\mu}+v_{1}\omega^{\mu-1}+\dots +v_{\mu} \qquad eqn12$$ Cantor added a ‘principle of limitation’ meant to impose an order of sorts upon the seemingly endless hierarchy of transfinite ordinal numbers.This made it possible to identify natural breaks in the sequence and to distinguish between the first denumerably infinite number class of natural numbers (I), the second number-class (II), and successively higher classes of transfinite ordinal numbers. The collection of all numbers $\alpha$(growing in definite succession) that may be created using the two generational principles is what we refer to as the second number-class (II). $$\omega, \omega+1, \dots ,v_{0}\omega^{\mu}+v_{1}\omega^{\mu-1}+\dots +v_{\mu}, \dots ,\omega^{\omega}, \dots, \alpha , \dots \qquad eqn13$$ One goal of Cantor’s transfinite set theory was to provide a means of resolving the hypothesis, that the set of real numbers $R$ was the next in power following the set of natural numbers $N$. Despite having high hopes that the transfinite ordinal numbers and other differences between various types of point sets will soon contribute to the resolution of the Continuum Hypothesis, none were found. Cantor was quite irritated by his inability to demonstrate its accuracy despite exerting enormous effort. He believed he had discovered a proof in the beginning of 1884, but he afterwards felt he could really refute the theory. He finally saw that he had made absolutely no progress.

i. One-To-One Correspondence

Cantor provided various instances to demonstrate that the ability to count items is not required to determine if two sets are equinumerous. The number of items included in a set is referred to as its cardinality or power. If two subsets have the same cardinal number, they belong to the same equivalence class.Cantor's definition stated that that two sets are equivalent sets if and only if every element of each one of them corresponds one and only one element of the other. [13]

ii. Different sizes of infinity

You could assume that since the sets of real numbers and natural numbers are both infinite, they both have the same size. However, this is a complete fallacy. Real numbers actually outnumber natural numbers by a large margin, and there is no method to organize the reals and the naturals so that we are assigning exactly one real number to each natural number. We'll use the contradiction technique to demonstrate this. First, we'll assume that the inverse of our claim is correct: that the real numbers are countably infinite, and so there is a method to line up all the reals with the naturals in a one-to-one correspondence. It doesn't matter how this correspondence appears, so let's assume the first few pairs in the correspondence are as follows:

The main premise here is that every single real number exists somewhere on this list. By creating a new number that does not appear in the list, we will prove that this is in fact incorrect. By altering the first decimal place in the first number, the second decimal place in the second number, and so on:

Now put all those changed numbers together: $0.2987\dots$. This new "diagonal" number is unquestionably real. However, it differs from the other numbers on the list in the following ways: its first digit differs from the first digit of our first number, its second digit differs from the second digit of our second number, and so on. We created a new real number that does not appear on our list. This runs counter to our central premise that every real number exists somewhere in the correspondence. As a result, the set of Real numbers is larger than the set of Natural numbers.

iii. The reception of the mathematics community to Cantor's studies and discoveries

Cantor's remarkable work and the fact that he dedicated his life to the study of infinities were initially disregarded, scorned, and rejected by the mathematical community at this time, which put him in a depressed state and required him to spend his final days commuting between mental facilities. Cantor's set theory will be regarded by future generations as "a disease from which one has recovered," according to Henri Poincare. He held that “most of the ideas of Cantorian set theory should be banished from mathematics once and for all.” Set theory, according to Hermann Weyl, a brilliant and versatile mathematician of the twentieth century, is a "house built on sand." [14] Cantor's work is now widely acknowledged in the philosophical and mathematical communities. Nearly every fundamental mathematical idea has some connection to Cantor, whether directly or indirectly. In mathematics, we can be certain of that.

IV. How do we generate thoughts of infinity?

i. The concept

It is difficult to find a cohesive definition of empirical ideas and abstract concepts. Most empirical notions are derived from an individual's observation and are designed to correspond to concrete things, making them comparatively easy to understand for the mind. Abstract conceptions, on the other hand, are regarded differently; they are not the result of a person's perceptions, and their aim is to relate to abstract entities. If an item is not embedded or is embedded in a network of tangential connections, it is abstract. The interaction between abstract things and the human mind is more intricate and interesting. The most significant distinction is how the mind perceives this idea. Some requirements must be met before a person may acquire an abstract notion, such as his mind having constructed something mentally referred to by this concept. The human mind acquires a notion as a consequence of a cognitive process that is influenced by aspects such as intuition, and knowledge.

ii. The acquisition of conceptions of infinity according to Cantor

Cantor was aware of the difficulty in gaining a grasp of the concept of infinity. There are two ways to consider things: consecutively or simultaneously. The successive technique pertains to the concept of numbers, but the simultaneous method refers to the concept of sets, and the human mind's capabilities allow it to detect sets, whether natural or abstract. With finite means and experiences, we cannot consider infinity. The experience of having no boundaries is necessary to comprehend the idea of potential infinite since it is possible to imagine that there is a bigger number for each given number. But how can we understand the idea of actual infinity? Since potential infinite is onto-logically reliant on actual infinity, potential infinity depends on actual infinity intellectually. Therefore, one cannot come to understand the idea of actual infinite via experiencing potential infinity. On the other hand, understanding the idea of actual infinite is required to understand the idea of potential infinity. Cantor argues that God instilled the concept of number, both finite and transfinite, into the mind of man, he wrote: "sowohl getrennt als auch in ihrer aktual unendlichen Totalit¨at als ewige Ideen in intellectu Divino im h¨ochsten Grad der Realit¨at existieren." [15] This suggests that transfinite numbers exist in man's mind just as everlasting thoughts exist in God's mind. However, the question how an individual human mind gets concept of infinity is still an unanswered question of cognitive science.

V. Conclusion

The concept of infinity was one of the most perplexing notions from the beginning of humanity, and many paradoxes were created to demolish it, but cantor was able to challenge all of them and show the existence of infinite mathematically and philosophically. However, this does not negate the truth that grasping the concept of infinity is still difficult for the human mind and occasionally goes beyond its capabilities. However, once we acquire this concept, we will be able to see how the human mind is capable of creating things that are larger than the universe itself.

I. Introduction

Elliptic Curves are smooth algebraic curves with the property that given any two points \(P_{1}\) and \(P_{2}\) on the curve you can always construct a 3^rd point on the curve. This method of producing new points on the curve is known as Diophantus method. A famous application to Diophantus method is finding a triangle with rational sides which has an area equal to 5 [3] . One can construct an equation of an elliptic curve with variables \(x\) and \(y\) related to the legs of a given right triangle. Thus, when a specific triangle with rational sides is found, infinitely other similar triangles can be found. Elliptic curve cryptosytems were introduced by Miller [1] and Koblitz [2] in the mid-1980s. Elliptic Curve cryptosystems offered the advantage of replacing RSA cryptosystems with equivalent elliptic curve cryptosystems with much smaller key sizes [7] . In a performance analysis done by Mahto, D., Yadav, D.K. [8] on RSA and ECC , it was found that elliptic curves are much superior than RSA in terms of encryption and decryption time, resources management and key sizes. They found that RSA based cryptosystems that used 1024/2048/3072-bit keys can be readily replaced by equivalent ECC with only 160/224/256-bit keys. Elliptic Curves used in cryptosystems are defined over finite fields. The points on an elliptic curve form an abelian group. Before investigating the group structure of elliptic curves over finite fields and defining the operation on the elements of the group, it’s useful to look at some of the properties of elliptic curves over \(\mathbb{R}\) and the geometry behind it to motivate the group law for addition of points. Then, it will be found that many of the results can be easily transferred over to elliptic curves over finite fields.

II. Elliptic Curves over \(\mathbb{R}\)

An Elliptic Curve \(E\) can be represented as an equation in the following form:$$E: y^2=x^3+Ax+B$$

where the coefficients \(A\) and \(B\) belong the to the field \(\mathbb{K}\) (with \(Char(\mathbb{K})>3)\) over which the curve \(E\) is defined. This equation will be referred to as Weierstrass equation.

Definition 2.1. An Elliptic Curve \(E\) over Field \(\mathbb{K}\) (with \(Char(\mathbb{K})>3\)) is denoted by \(E(\mathbb{K})\). \(E\) consists of all points such that with coefficients \(A\) and \(B \in \mathbb{K}\): $$E(\mathbb{K}): \{\infty\} \cup \{(x,y) \in \mathbb{K} \times \mathbb{K} \mid y^2=x^3+Ax+B\}$$ Where, \(\infty\) is called the point at infinity and will be the additive identity for the group of points on the elliptic curve.

For this section, we will deal with the case when \(\mathbb{K}\) is \(\mathbb{R}\). It’s required that the elliptic curve is non-singular, i.e. it doesn’t have repeated roots. A singular curve with a root of multiplicity \(2\) will intersect itself, as shown in Figure 1. The curve \(y^2=x^3+5x^2\) has a double root at \(x=0\). A singular curve that has a root with multiplicity \(3\) (such as \(y^2=x^3\)) will have a cusp, as shown in Figure 2. If the elliptic curve used in a cryptosystem has repeated roots, this will result in a weak cryptosystem that can be broken easily.

What conditions on an algebraic curve \(f(x,y)=0\) must be fulfilled to ensure that it’s non-singular? Singular points on a curve can be thought as having a gradient that’s not well-defined: The gradient vanishes at these points [4].

Definition 2.2. A point \(P\) on an algebraic curve \(f(x,y)=0\) is singular if \(\nabla f(P) = 0\). A curve is called non-singular if it has no singularities.

Theorem 2.1. [7] The elliptic curve $$E: y^2=x^3+Ax+B$$ over \(\mathbb{R}\), where \(A, B >0\), is non-singular if and only if \(4A^3+27B^2 \neq 0\)

Proof. An elliptic curve of the form \(f(x,y)=y^2-x^3-Ax-B=0\) will be singular if: $$\nabla f = \begin{pmatrix} -3x^2-A\\ 2y \end{pmatrix}=\begin{pmatrix} 0\\ 0 \end{pmatrix}$$ Thus, the point \((\frac{i\sqrt{3A}}{3},0)\) is a singular point. Substituting this back to the equation of the curve: $$\frac{iA\sqrt{3A}}{9}-\frac{Ai\sqrt{3A}}{3}-B=0 \rightarrow \frac{-4A^3}{27}=B^2$$ Thus, an elliptic curve will have a singular point if and only if $$4A^3+27B^2=0$$

III. Basic Group theory

A group is a set of elements accompanied by a binary operation with certain properties fulfilled. Examples of groups include \(\mathbb{Z}\) which is an infinite group with the binary operation addition. \(\mathbb{R}\) is another infinite group under either the addition or multiplication operation. Points on an elliptic curve also form a group. So, to understand the structure of the points on an elliptic curve, it's important to study some major results in group theory.

i. Groups and binary operations

Before we exactly define what is a group, we will define what is exactly a binary operation on a set \(S\).

Definition 3.1. A binary operation \(*\) on a set \(S\) is a function mapping \(S \times S\) into \(S\). For each \((a, b) \in S \times S\), we will denote the element \(*((a,b))\) of \(S\) by \(a*b\)

Example 3.1. Addition of numbers is a binary operation \(+\) on the set of rational numbers \(\mathbb{Q}\). We can define another binary operation \(*\) on the set of rational numbers \(\mathbb{Q}\) as follows: $$\text{For } a \text{ and } b \in \mathbb{Q}: a*b = \frac{ab}{2}$$ Thus, the binary operation \(*\) maps every pair of elements \((a,b) \in \mathbb{Q} \times \mathbb{Q}\) into the element \(\frac{ab}{2} \in \mathbb{Q}\).

Sets equipped with a binary operation that’s associative, with an identity element, and inverses for each element are called groups.

Definition 3.2. A group \(\langle G,* \rangle\) is a set closed under the binary operation \(*\), satisfying the following axioms:

1. \(G\) is associative: $$\forall a,b,c \in G, (a*b)*c =a*(b*c)$$
2. \(G\) has an identity element: $$\forall a \in G, \exists! e \in G: a*e =e*a=a$$
3. \(G\) has an identity element: $$\forall a \in G, \exists! a^\prime: a*a^\prime = a^\prime * a = e$$

Groups that are commutative are called abelian groups.

Definition 3.3. A group \(\langle G,* \rangle\) is called an abelian group if the binary operation \(*\) is commutative.

Example 3.2. Consider \(\langle \mathbb{Z}_n,+ \rangle\). \(0\) is the identity element and the inverse of a number \(a\) is \(n-a\). The addition of numbers modulo \(n\) is associative. Thus, \(\langle \mathbb{Z}_n,+ \rangle\) is a group. Furthermore, since addition is commutative, it’s an abelian group.

Example 3.3. The set of all one-to-one functions under the operation of composition is a group. It's associative: $$(f \circ g) \circ h = f \circ (g \circ h)$$ The function \(g(x)=x\) is the identity element: $$(f \circ x)(x)=f(x)$$ Every one-to-one function has an inverse \(f^{-1}\): $$(f \circ f^{-1})(x)=(f^{-1} \circ f)(x)=x$$ However, it's not commutative: $$f \circ g \neq g \circ f$$

ii. Groups Isomorphism

Isomorphisms between groups are one-to-one and onto mappings that preserve the structure of one group while mapping it into the other. Take for example the group \(\langle\{e,a\},*\rangle\), where \(a*a=e\). The group table for this group is shown in Table \(1\).

It can be checked that this 2-element group satisfies all the axioms of a group (associativity, existence of an identity element, and existence of an inverse). Another 2-element group is the group \(\{1,-1\}\) under multiplication \((\langle\{1,-1\}, \cdot \rangle)\). Its group table is shown in Table 2. We see that there's a similarity between the two group tables. In fact, if we relabel \(1\) by \(e\) and \(-1\) by \(a\), we get the first group table. If there's a one-to-one and onto labeling of the elements of one group to match elements of the other group while preserving the group structure, then we say that both groups are isomorphic.

Definition 3.4. Let \(\langle G_1, *_1 \rangle\) and \(\langle G_2, *_2 \rangle\) be groups and \(f: G_1 \rightarrow G_2\) . We say that \(f\) is a group isomorphism if the following two conditions are satisfied:

1. The function \(f\) is one-to-one and maps onto \(G_2\)
2. For all \(a,b \in G_1\) ,\(f(a *_1 b) = f(a) *_2 f(b)\).

Example 3.4. The group of integers \(\mathbb{Z}\) and even integers \(2\mathbb{Z}\) are isomorphic with both groups under the binary operation of addition. We can set-up a function \(f\) such that: $$f: \mathbb{Z} \rightarrow 2\mathbb{Z}$$ $$f(x)=2x$$ This function is one-to-one and onto. Furthermore: $$f(x+y)=2(x+y)=2x+2y=f(x)+f(y)$$ Thus, The group of integers \(\mathbb{Z}\) and even integers \(2\mathbb{Z}\) are isomorphic: $$\mathbb{Z} \simeq 2\mathbb{Z}$$

Isomorphisms between groups can provide a way to solve problems that may be hard to attack in one group but trivial to solve in the other group. One such situation is when trying to solve the discrete logarithm problem for singular elliptic curves. Instead of working with points on the curve, one can instead work with elements from the field over which the elliptic curve is defined, for example.

iii. Cyclic Groups and Subgroups

If every element in a group can be represented as some power (or multiple) of a some fixed element, then this group is called cyclic. The element that "generates" the group is therefore called a generator. In another way, let \(n\) be the order (number of elements) of the the cyclic group \(G\) with generator \(a\), then $n$ is the smallest number such that: $$a^n = 1$$ The order of an element \(b \in G\), where \(G\) is a cyclic group, is the smallest integer \(x\) such that: $$b^x = 1$$

Theorem 3.1. [10] All cyclic groups are isomorphic.

Proof. Let \(G\) be a cyclic group generated by \(a\): $$G = \{a,a^2,a^3,...,a^n=e\}$$ Also, let \(G^\prime\) be another cyclic group generated by \(b\) with the same order as \(G\): $$G^\prime = \{b,b^2,b^3,...,b^n=e\}$$ Then we claim that the mapping \(\psi: G \rightarrow G^\prime\) defined by \(\psi(a^s) = b^s\) is an isomorphism. First of all, it's one-to-one (Assume \(\psi(a^s)=\psi(a^r)\) for \(s \neq r\) and \(n,r < n\), then \(b^s=b^r\) which is a contradiction). Also, the map is onto. Therefore, \(\psi\) is a bijection from $G$ to \(G^\prime\). The map \(\psi\) satisfies the homomorphism property: $$\psi(a^s \cdot a^r)=\psi(a^{s+r})=b^{s+r}=b^s \cdot b^r=\psi(a^s) \cdot \psi(a^r)$$ Thus, all cyclic groups of the same order are isomorphic.

Corollary 3.1. All cyclic groups of order \(n\) are isomorphic to \(\langle \mathbb{Z}_n,+_n \rangle\)

Theorem 3.2. [10] Let \(H\) be a subgroup of the cyclic group \(G\). Then, $H$ is also cyclic.

Proof. Assume that \(H\) contains elements other than the identity element \(e\) and let \(G\) be generated by \(a\). Let \(x \in \mathbb{Z}^{+}\) be the smallest integer such that \(a^x \in H\). We will prove that \(a^x\) generates \(H\). In other words we must show that if \(b \in H\), then \(b = (a^x)^y = a^{xy}\) for \(y \in \mathbb{Z}^{+}\). Since \(b \in G\), we have \(b = a^n\) for some \(n \in \mathbb{Z}^{+}\). Now we want to show that \(n=xy\) or that \(y \mid n\). Using the Euclidean algorithm we have: $$n = qx+r \hspace{0.5 cm} \text{for} \hspace{0.5cm} 0 \leq r < x$$ Therefore: $$b=a^n=a^{qx+r}=(a^x)^qa^r$$ Now since \(a^x \in H\), then \((a^x)^{-q} \in H\). Therefore: $$a^n a^{-qx} \in H \rightarrow a^r \in H$$ But \(r < x\), and \(x\) was assumed to be the smallest positive integer such that: $$a^x \in H$$ Therefore, \(r=0\): $$n=qx \hspace{0.5cm} \text{For some } q \in \mathbb{Z}^{+}$$ $$b=a^n=a^{qx}=(a^x)^q$$ Thus, any element \(b \in H\) can be generated by \(a^x\) and \(H\) is a cyclic subgroup of \(G\).

Let \(b \in G\), where \(G\) is a cyclic group. How many elements does the cyclic group \(H\) generated by \(b\) have? Also, how many subgroups does a cyclic group of order \(n\) have?

Theorem 3.3. [10] Let \(G\) be a cyclic group of order \(n\), generated by the element \(a\). Let \(b \in G\) and let \(b = a^{m}\). Then \(b\) generates a cyclic subgroup \(H\) of \(G\) containing \(\frac{n}{gcd(m,n)}\) elements.

Proof. Since \(H\) is generated by \(b = a^m\), then the order of \(H\) is the smallest number \(s\), such that \(b^s=a^{ms}=e\). But since \(n\) is the smallest positive integer such that \(a^n = e\), then: $$n \mid ms$$ The smallest value of \(s\), such that \(n \mid ms\), is the order of \(H\). Let \(d = gcd(m,n)\), then \(gcd(m/d,n/d)=1\). Thus \(s\) is the smallest value such that: $$\frac{ms}{n} = \frac{s(m/d)}{(n/d)} \hspace{0.2cm} \text{is an integer.}$$ But since \(gcd(m/d,n/d)=1\), then \(\frac{n}{d} \mid s\). Thus, the smallest value of \(s\) is: $$s = |H| = \frac{n}{d} = \frac{n}{gcd(m,n)}$$

Theorem 3.4. [10] Let \(G\) be a cyclic group of order \(n\). If the prime factorization of \(n\) is: $$n=p_{1}^{k_1}p_{2}^{k_2}...p_{m}^{k_m}$$ Then, the number of cyclic subgroups of \(G\) is: $$(k_1+1)(k_2+1)...(k_m+1)$$

Proof. By theorem 5.2, the order of any cyclic subgroup \(H\) of the cyclic group \(G\) divides \(n\). Therefore, the number of subgroups of the group \(G\) is the number of divisors of \(n\) which is: $$(k_1+1)(k_2+1)...(k_m+1)$$

Cyclic groups are not the only kind of groups in which its subgroups' order divides the group's order. In fact, the order of a subgroup of any finite group divides the group order as stated by Theorem of Lagrange.

Theorem 3.5 (Theorem of Lagrange). [10] Let \(H\) be a subgroup of a finite group \(G\). Then the order of \(H\) is a divisor of the order of \(G\).

iv. Finite Abelian groups

Finite groups can be combined to from another groups. There are two ways to do so, either by the direct product of the groups \(G_i\) for all \(i\) if the binary operation of the groups is multiplicative, or by the direct sum of the groups if their binary operation is additive.

Definition 3.5. The direct sum of the sets \(S_1,S_2,...,S_n\) is the set of all ordered \(n-tuples\) \((s_1,s_2,...,s_n)\), where \(s_i\in S_i\). The direct sum is denoted by either $$S_1 \oplus S_2 \oplus ... \oplus S_n$$ or by $$\bigoplus_{i=1}^{n}S_i$$

Theorem 3.6. Let \(S_1,S_2,··· ,S_n\) be groups. For \((a_1,a_2 ,...,a_n)\) and \((b_1,b_2,...,b_n)\) in \(\oplus_{i=1}^{n}S_i\) define \((a_1,a_2 ,...,a_n)+(b_1,b_2,...,b_n)\) to be the element \((a_1+b_1,a_2+b_2 ,...,a_n+b_n)\). Then, \(\oplus_{i=1}^{n}S_i\) is a group, the direct sum of the groups \(S_i\), under this binary operation.

Theorem 3.7. [10] [3] A finite abelian group \(G\) is ismorphic to a group of the form: $$\mathbb{Z}_{n_1} \oplus \mathbb{Z}_{n_2} \oplus ... \oplus \mathbb{Z}_{n_r}$$ where \(n_i\mid n_{i+1}\) for \(1\leq i \leq r-1\).

IV. Group Law for Elliptic Curves over $\mathbb{R}$

As mentioned, the importance of elliptic curves arises from the fact that the combination of two points can produce another point on the curve. This means that the elliptic curve is closed under this operation of combination of points. This operation will be called addition. To add two points $P_{1}$ and $P_2$ on an elliptic curve, draw a line $L$ through the two points. The line $L$ will intersect the curve $E$ again at another point $N$ with coordinates $(x,y)$, as shown in Figure 3. Reflect this point across the $x$-axis to get the point $P_3 = (x,-y)$. This operation will be called the addition of two points $P_1$ and $P_2$ such that: $$P_1+P_2=P_3$$ An addition of a point $P$ to itself can be defined in a similar manner. Draw the tangent $T$ to the point $P$. It will intersect the curve $E$ at another point $M=(x,y)$, as shown in Figure 4. Again, reflect this point across the $x-axis$ to get point $Q$ such that: $$2P=Q$$

Suppose we have the elliptic curve $E: y^2=x^3+Ax+B$ and we want to add the two points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$. The slope of the line through the two points is: $$m=\frac{y_2-y_1}{x_2-x_1}$$ The equation of the line $L$ through the two points is: $$y=m(x-x_1)+y_1$$ This line intersects with the elliptic curve in $3$ points: $P_1,P_2,$ and the other point $N$ we are looking for. So, we substitute back into the equation of the curve to get: $$(m(x-x_1)+y_1)^2=x^3+Ax+B$$ We rearrange this equation to get: $$x^3-m^2x^2+...=0$$ We only care about the coefficient of $x^2$, since it's the sum of the roots of the equation,i.e. it's the negative of the sum of the x-coordinates of the $3$ points $P_1,P_2,$ and $N=(x,y)$. Thus $$x_N=m^2-x_1-x_2$$ $$y_N=m(x_N-x_1)-y_1$$ Then, reflect the point $N=(x_N,y_N)$ across the x-axis to get $P_3=P_1+P_2=(x_3,y_3)$: $$x_3=m^2-x_1-x_2$$ $$y_3=m(x_1-x_3)-y_1$$ The point at infinity $\infty$ acts as an additive identity. Lines through infinity are vertical, so the line through a point $P$ and $\infty$ is vertical and intersects the curve only once again at another point, namely the reflection of $P$ across the x-axis. Therefore, when this point is reflected across the x-axis to get the point $$P+\infty$$ It gives back $P$. Thus $$P+\infty=P$$ Now remains the case of adding one point to itself. The steps of deriving the coordinates for the point $2P$ is the same as above but with $x_1=x_2$ and $y_1=y_2$. The slope is just the slope of the tangent line to the curve at this point: $$2y\frac{dy}{dx}=3x^2+A \rightarrow m = \frac{3x^2+A}{2y}$$ Now, we are ready to define a group law for the set of points on an elliptic curve:

Definition 4.1 (Group Law). [3] Consider the elliptic curve $E: y^2=x^3+Ax+B$. Let $P_1=(x_1,y_1)$ and $P_2=(x_1,y_2)$. Define $P_3=(x_3,y_3)$ Define a binary operation, or a group law, on points on this curve such that $P_1+P_2=P_3.$:

1. If $x_1 \neq x_2$ then: $$x_3=m^2-x_1-x_2 \hspace{1cm} y_3=m(x_1-x_3)-y_1 \hspace{1cm} \text{where } m = \frac{y_2-y_1}{x_2-x_1}$$
2. If $x_1 = x_2$ but $y_1 \neq y_2$ then: $$P_1+P_2=\infty$$
3. If $P_1=P_2$ and $y_1 \neq 0$ then: $$x_3=m^2-2x_1 \hspace{1cm} y_3=m(x_1-x_3)-y_1 \hspace{1cm} \text{where } m = \frac{3x^2+A}{2y}$$
4. If $P_1=P_2$ and $y_1 = 0$, then: $$P_1+P_2=\infty$$
5. The point $\infty$ the an identity element where for any $P \in E$: $$P+\infty = P$$
6. If $P_1+P_2=\infty$, then $P_2$ is the inverse of $P_1$: $$P_2=(x_2,y_2)=-P_1=(x_1,-y_1)$$

The of points of an elliptic curve over a finite field form a group. The group law is a binary operation on points on an elliptic curve. It maps each pair of points $(P_1,P_2)$ on the curve to a third point $P_3$ through the group law. It can then proved that points on the elliptic curve form a group under the binary operation of points addition.

Theorem 4.1. [3] [4] The set of points on a non-singular elliptic curve $E$ form an abelian group under the group law of addition.

Proof. The proof follows from the definition of the group law. This will be proved for elliptic curves of $Char(\mathbb{K})>3$ but it can be proved in the same way for elliptic curves defined over finite fields with $Char(\mathbb{K})=3$ or $Char(\mathbb{K})=2$.

1. The group law is associative [5]: $$\forall A, B, C \in E, (A+B)+C=A+(B+C)$$
2. >Existence of an identity element: $$\forall A \in E, A + \infty = \infty + A = A$$
3. Existence of inverses: $$\forall A=(x,y) \in E, \exists! A^\prime = (x,-y): A+A^\prime = \infty$$
4. The group of points is abelian:

   The line $L$ from point $P_1$ to $P_2$ is the same as the line from $P_2$ to $P_1$ and intersects the curve $E$ at same third point $P_3$ such that: $$P_1+P_2=P_2+P_1=P_3$$

V. Elliptic Curves over Prime Fields $\mathbb{Z}_{p}$

The group law for elliptic curves over finite fields is the same as defined for elliptic curves over $\mathbb{R}$. But all calculations must be carried out over the field and inverses for elements of a finite field must be found to compute the fraction that arises when trying to compute slope of the line connecting two points on the curve or the tangent line to a point.

Example 5.1. Consider the following elliptic curve defined over $\mathbb{Z}_{7}$ $$E(\mathbb{Z}_{7}): y^2=x^3+5x+5$$ We want to find the points on $E$. Therefore, $x$ is allowed to run through all the values of the field $\mathbb{Z}_{7}$: $$x \equiv 0 \rightarrow \text{No solutions} \mod{7} \hspace{2cm} x \equiv 1 \rightarrow y \equiv 2,5 \mod{7}$$ $$x \equiv 2 \rightarrow y \equiv 3,4 \mod{7} \hspace{2cm} x \equiv 3 \rightarrow \text{No solutions} \mod{7}$$ $$x \equiv 4 \rightarrow \text{No solutions} \mod{7} \hspace{2cm} x \equiv 5 \rightarrow y \equiv 1,6 \mod{7}$$ $$ x = \infty \rightarrow y = \infty$$ Thus, the elliptic curve $E(\mathbb{Z}_{7})$ consists of the following points: $$\{\infty,(1,2),(1,5),(2,3),(2,4),(5,1),(5,6)\}$$

Example 5.2. For the elliptic curve defined in the previous example, we will try to add the points $(1,2)$ and $(5,1)$: $$m \equiv (y_2-y_1)\cdot (x_2-x_1)^{-1} \equiv (1-2) \cdot (5-1)^{-1} \equiv 6 \cdot 2 \equiv 5 \mod{7}$$ $$x_3 \equiv m^2-x_1-x_2 \equiv 5^2 - 1 - 5 \equiv 5 \hspace{2cm} y_3 \equiv m(x_1-x_3)-y_1 \equiv 5(1-5)-2 \equiv 6$$ Therefore: $$(1,2)+(5,1) \equiv (5,6)$$

i. Elliptic Curves over fields of $Char(2)$:

The equations developed for the group law on an elliptic curve don’t work on elliptic curves of $Char(2)$. This is for a simple reason, the equation used for such curves are different from Weierstrass equation. If we take the derivative of Weierstrass equation, we have: $$\frac{d}{dx}y^2 \equiv 2yy^{\prime} \equiv 0$$ since $2 \equiv 0$ in fields of $Char(2)$. Therefore, a modified version of Weierstrass equation is used. The generalized Weierstrass equation is of the following form: $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ The process followed in the addition of points over $\mathbb{R}$ is the same here over $Char(2)$ but the reflection of a point across the $x-axis$ is not just a mere change of sign for the $y-coordinate$. To find the “reflection” of a point $P_1=(x_1,y_1)$, a point $P_2=(x_2,y_2)$ must be found such that: $$P_1+P_2 = \infty$$

Theorem 5.1. [3] The inverse of the point $P_1=(x_1,y_1)$ on the curve $E(\mathbb{K}_{2^q})$ defined by The generalized Weierstrass equation: $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ is the point $P_2$ with the following coordinates: $$-(x_{1},y_{1})=(x_{1},-a_{1}x-a_{3}-y_{1})$$

Proof. The points $P_1$ and $P_2$ have the following property: $$P_1 + P_2 = \infty$$ Therefore, they have the same $x-coordinate = x_1$ since a line through $P_1, P_2$ and $\infty$ is just a vertical line. Thus: $$(x_2,y_2)=(x_1,y_2)$$ Plugging the coordinates of both points $P_1$ and $P_2$, we have: $$ x^{3}_{1}+a_2 x^{2}_{1}+a_4x_1+a_6= x^{3}_{2}+a_2 x^{2}_{2}+a_4x_2+a_6$$ Therefore, $$ y^{2}_{2}+a_1x_1y_2+a_3y_2= y^{2}_{1}+a_1x_1y_1+a_3y_1 $$ $$( y^{2}_{2}- y^{2}_{1})+ a_1x_1(y_2-y_1)+a_3(y_2-y_1)=0$$ dividing both sides by $y_2-y_1$: $$y_{2}+ y_{1}+ a_1x_1+a_3=0 \rightarrow y_{2}= -a_{1}x_{1}-a_{3}-y_{1} $$

ii. Singular Curves

The curves we have dealt assumed that the expression $x^3+Ax+B$ had distinct $3$ roots. These curves are the curves used in ECC. Singular elliptic curves over a field $\mathbb{K}$ are elliptic curves where the expression $x^3+Ax+B$ has multiple roots. This interesting case of elliptic curves results in an isomorphism between the non-singular points on the elliptic curve and the additive group $\mathbb{K}$ in the case of triple root, or isomorphic to the multiplicative group $\mathbb{K}^\times$ in the case of double root.

iii. Singular Elliptic Curves with triple roots

Example 5.3. Consider the elliptic curve $$E: y^2 = x^3$$ This curve has triple root at $x=0$. It's impossible to add this point to any other point on the elliptic curve since any line through the point $(0,0)$ passes through at most only one other point on the curve. However, the non-singular points on the curve $E$ form a group which we will denote by $E_{ns}(\mathbb{K})$ with the group law defined on it the same way as for non-singular elliptic curves.

Theorem 5.2. [3] Let $E_{ns}(\mathbb{K})$ denote the non-singular points on the elliptic curve $y^2=x^3$, defined over the finite field $\mathbb{K}$. Consider the following map: $$\psi: (x,y) \rightarrow \frac{x}{y}, \hspace{0.5cm} \infty \rightarrow 0$$ It's a group isomorphism between $E_{ns}(\mathbb{K})$ and the additive group $\mathbb{K}$

Proof. The proof will be divided into two parts: The first part will prove that the map $\psi$ is a bijection. The second part will prove that $\psi$ is a homomorphism. Let $t = \frac{x}{y}$. From the elliptic curve equation we have: $$x = \frac{y^2}{x^2} = \frac{1}{t^2}, \hspace{1cm} y = \frac{x}{t}=\frac{1}{t^3}$$ From these relations, it's evident that given a $(x,y)$, a unique $t$ can be found. Also, given a $t$, the corresponding $(x,y)$ can be found. Thus, the map $\psi$ is a bijection. Now, we must prove that $\psi$ is a homomorphism: $$\psi((x_1,y_1)+(x_2,y_2))=\psi(x_3,y_3)=\psi(x_1,y_1)+\psi(x_2,y_2)=t_1+t_2=t_3$$ By the group law, we have: $$x_3 = \left(\frac{y_2-y_1}{x_2-x_1}\right)^2-x_2-x_1$$ Using the fact that $x_i=1/t_{i}^2$ and $y_i=1/t_{i}^3$, we have: $$t_{3}^{-2} = \left(\frac{t_{2}^{-3}-t_{1}^{-3}}{t_{2}^{-2}-t_{1}^{-2}}\right)^2-t_{2}^{-2}-t_{1}^{-2}$$ A simplification yields: $$t_{3}^{-2}=(t_{1}+t_{2})^{-2}$$ Thus: $$t_1+t_2=t_3$$ Finally, this proves that the map $\psi$ is an isomorphism and that $E_{ns}(\mathbb{K})$ is isomorphic to the additive group $\mathbb{K}$.

iv. Singular Elliptic Curves with double roots

For the double root case, an isomorphism also exists between $E_{ns}(\mathbb{K})$ and the multiplicative group $\mathbb{K}^\times$. Consider the singular elliptic curve $E: y^2=x^2(x+a)$. The only singular point on this curve is the point $(0,0)$. Now, let $\beta^2 = a$. $\beta$ might not exist in $\mathbb{K}$ but exists in an extension of $\mathbb{K}$.

Theorem 5.3. [3] Let $E_{ns}(\mathbb{K})$ denote the non-singular points on the elliptic curve $y^2=x^2(x+a)$, $a \neq 0$ defined over the finite field $\mathbb{K}$. Let $\beta^2 = a$. Consider the following map: $$\psi: (x,y) \rightarrow \frac{y+\beta x}{y-\beta x}, \hspace{0.5cm} \infty \rightarrow 1$$

1. If $\beta \in \mathbb{K}$, then $\psi$ is an isomorphism from the group $E_{ns}(\mathbb{K})$ to the multiplicative group $\mathbb{K}^\times$.
2. If $\beta \not\in \mathbb{K}$, then $\psi$ is an isomorphism: $$E_{ns}(\mathbb{K}) \simeq \{m+\beta n \mid m,n \in \mathbb{K}, m^2-an^2=1\}$$ which is also a multiplicative group.

Proof. We will prove that $\psi$ is a bijection in each case separately: Assume $\beta \in \mathbb{K}$. Let $t = \frac{y+\beta x}{y - \beta x}$, then by solving for $\frac{y}{x}$ and using the fact that $x+a = \frac{y^2}{x^2}$, we have: $$x = \frac{4\beta^2 t}{(t-1)^2}, \hspace{0.5cm} y = \frac{4 \beta^3t(t+1)}{(t-1)^3}$$ This means that given a $t$, one can find a unique $(x,y)$. Also, given a $(x,y)$, a unique $t$ can be found. Therefore, the mapping $\psi$ is a bijection.\\ Now, assume that $\beta \not \in \mathbb{K}$. Rationalize the denominator for $t$: $$\frac{y+\beta x}{y-\beta x}=\frac{(y+\beta x)^2}{y^2-ax^2} = \frac{y^2+2\beta x +ax^2}{x^3} = \frac{y^2+ax^2}{x^3}+\beta \frac{2}{x^2} = m+\beta n$$ Similarly: $$\frac{y-\beta x}{y+\beta x} = m-\beta n$$ Multiply these two expressions: $$m^2-an^2= (m+\beta n)(m-\beta n) =\frac{y+\beta x}{y-\beta x} \cdot \frac{y-\beta x}{y+\beta x} = 1 $$ Therefore, given a $(x,y) \in E_{ns}(\mathbb{K}), \exists! m,n \in \mathbb{K}: m-an^2=1$. We will prove the converse now to prove that $\psi$ is a bijection. Let: $$ x = (\frac{u+1}{v})^2-a, \hspace{0.5cm} y = (\frac{u+1}{v})x$$ It can be verified indeed that the pair $(x,y) \in E_{ns}(\mathbb{K})$. Furthermore, $\psi(x,y) = m+\beta n$. Therefore it can be deduced that the mapping $\psi$ is indeed a bijection. Now we will prove that $\psi$ is a homomorphism. In other words, given $(x_1,y_1)+(x_2,y_2)=(x_3,y_3)$, we must show that $t_1t_2=t_3$.
By the group law, we have: $$x_3 = (\frac{y_2-y_1}{x_2-x_1})^2-a-x_2-x_1$$ Now, substituting $x_k = \frac{4 \beta^2 t_k}{(t_k-1)^2}$ and $y_k = \frac{4 \beta^3 t_k(t_k+1)}{(t_k-1)^3}$ and simplifying the resulting algebraic expression: \begin{align} \frac{t_3}{(t_3-1)^2} &= \frac{t_1t_2}{(t_1t_2-1)^2} \end{align} Similarly, from the group law and the same substitution: $$y_3 = (\frac{y_2-y_1}{x_2-x_1})(x_1-x_3)-y_1$$ \begin{align} \frac{t_3(t_3+1)}{(t_3-1)^3} &= \frac{t_1t_2(t_1t_2+1)}{(t_1t_2-1)^3} \end{align} Taking the ratio of $(1)$ and $(2)$: $$\frac{t_3-1}{t_3+1} = \frac{t_1t_2-1}{t_1t_2+1}$$ Which simplifies to the following: $$t_1t_2=t_3$$ Thus, the mapping $\psi$ is an isomorphism as desired.

VI. Group structure of Elliptic Curves

In this section, we discuss the group properites of points on an elliptic curve as well as the torsion groups of elliptic curves. We also discuss Hasse's theorem, which is a siginficant result on the order of a group of points on an elliptic curve.

i. Torsion Points

Torsion points of a an elliptic curve are points which have a finite order; that is, there exists $m \in \mathbb{Z}^{+}$ such that $mP = \infty$ for any torsion point. All points on an elliptic curve are torsion points. In this section, the group structure of $n-torsion$ points (Points $P$ with $nP = \infty$) will be studied in the special cases of $n=2$ and $n=3$ and generalized for any $n$. Define an elliptic curve $E$ over a finite field $\mathbb{K}$. Then the $n-torsion$ points of $E$ are the elements of the following set: $$E[n]=\{P \in E(\overline{\mathbb{K}}) \mid nP= \infty\}$$

Theorem 6.1. [3] Let $E$ be an elliptic curve over a field $\mathbb{K}$. If the the characteristic of $\mathbb{K}$ is not $2$, then: $$E[2] \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_{2}$$ if the characteristic of $\mathbb{K}$ is 2, then: $$E[2] \simeq 0 \hspace{0.2cm} or \hspace{0.2cm} \mathbb{Z}_2$$

Proof. We will prove the case for $Char(\mathbb{K}) \neq 2$, and a similar analysis will yield the proof for the case when $Char(\mathbb{K})=2$. Then, the equation for the elliptic curve $E$ can be written in the following form: $$y^2 = (x-r_1)(x-r_2)(x-r_3)$$ Where $r_1,r_2,$ and $r_3$ all $\in$ the algebraically closed field $\overline{\mathbb{K}}$. Now, $E[2]$ consist of all points $P$ such that $2P = \infty$. This means that the tangent line at $P$ intersects the curve at infinity,i.e, it's a vertical line. Therefore, $y=0$, and we have the following points as $2-torsion$ points of the elliptic curve $E$: $$E[2] = \{\infty,(r_1,0),(r_2,0),(r_3,0)\}$$ Since every element in $E[2]$ has order $2$, and this group consists of $4$ elements, then $E[2]$ is isomorphic to $\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$. A similar analysis to elliptic curves over fields with characteristic 2 yields that $E[2]$ is isomorphic either to $0$ or $Z_2$ (depending on the equation used for defining the curve).

Theorem 6.2. [3] Let $E$ be an elliptic curve over a field $\mathbb{K}$. If the the characteristic of $\mathbb{K}$ is not $3$, then: $$E[3] \simeq \mathbb{Z}_3 \oplus \mathbb{Z}_{3}$$ if the characteristic of $\mathbb{K}$ is $3$, then: $$E[3] \simeq 0 \hspace{0.2cm} or \hspace{0.2cm} \mathbb{Z}_3$$

Proof. If the characteristic of the field is not $3$ or $2$, then the elliptic curve's equation can be written in the following form: $$y^2=x^3+Ax+B$$ Now, $E[3]$ consists of all points $P$ such that $3P = \infty \rightarrow 2P = -P$, which means that $2P$ and $P$ have the same $x-coordinate$. Using the group law for doubling a point we have: $$m^2=3x \hspace{1cm} y = \frac{3x^2+A}{2m}$$ Substituting into the elliptic curve equation we have: $$(3x^2+A)^2=12x(x^3+Ax+B)$$ Which simplifies to the following: $$3x^4+6Ax^2+12Bx-A^2=0$$ Since the discriminant of this polynomial, which is $-6912(4A^3+27B^2)^2$, is nonzero, then all the roots of the polynomial are distinct. Therefore, there are $4$ values for $x$ in $\overline{\mathbb{K}}$, and for each value there are $2$ values for $y$, with a total of $8$ points $P$ with $3P = \infty$. Therefore, including the point $\infty$, the group $E[3]$ has $9$ points in which every points is a $3-torsion$. Therefore, $E[3]$ is isomorphic to $Z_3 \oplus Z_3$. A similar analysis for curves defined over fields with characteristic $2$ yields that $E[3]$ is isomorphic to $\mathbb{Z}_3 \oplus \mathbb{Z}_3$. If the curve is defined over field of characteristic $3$, then it's isomorphic to $\mathbb{Z}_3$.

From these two results we can devise a general conclusion on the group structure of $E[n]$ for any $n$.

Theorem 6.3. [3] Let $E$ be an elliptic curve over a field $\mathbb{K}$ and let $n$ be a positive integer. If $Char(\mathbb{K}) \nmid n$ , or is $0$, then: $$E[n] \simeq \mathbb{Z}_{n} \oplus \mathbb{Z}_{n}$$ If $Char(\mathbb{K})=p$ and $p \mid n$, write $n = p^rn^\prime$ with $gcd(p,n^\prime)=1$. Then: $$E[n] \simeq \mathbb{Z}_{n^\prime} \oplus \mathbb{Z}_{n^\prime} \hspace{1cm} or \hspace{1cm} \mathbb{Z}_{n} \oplus \mathbb{Z}_{n^\prime}$$.

ii. Group structure of elliptic curves

Example 6.1. Lets consider the elliptic curve $y^2=x^3+x+1$ defined over the field $\mathbb{Z}_{5}$. Letting $x$ run through the $5$ values of $\mathbb{Z}_{5}$ and straightforward calculations yields that: $$E(\mathbb{Z}_{5}) = \{\infty,(0,1),(0,4),(2,1),(2,4),(3,1),(3,4),(4,2),(4,3)\}$$ So, $E(\mathbb{Z}_{5})$ has order $9$. It can further be shown that this is a cyclic group generated by the point $(0,1)$. By Theorem 5.5, the order of the subgroup generated by $(0,1)$ must divide $9$. Therefore either $3$ is the smallest positive integer such that $3(0,1)=\infty$, or $9$ is the smallest positive integer such that $9(0,1)=\infty$, which would mean that $E(\mathbb{Z}_{5})$ is cyclic. The group law can be used to obtain the point $P = 3(0,1)$. First, a calculation of $2(0,1)=(x,y)$ is performed: $$x \equiv m^2-2\cdot 0 \equiv 3^2 \equiv 9 \equiv 4 \hspace{1cm} y \equiv 3(0-4)-1 \equiv -13 \equiv 2 \mod{5}$$ Now, add this point to $(0,1)$ to get $3(0,1)$: $$(0,1)+(4,2)=3(0,1)=(2,1)$$ Therefore, the point $(0,1)$ can't have order $3$, and its order must be $9$ Therefore, $E(\mathbb{Z}_{5})$ is cyclic and generated by $(0,1)$. By Corollary 5.1, $E(\mathbb{Z}_{5})$ is isomorphic to $\mathbb{Z}_{9}$. Example 5.1 also provides another example of an elliptic curve that is isomorphic to a cyclic group. The elliptic curve has order $7$ and thus by theorem of Lagrange, the only possible value for the order of subgroups generated by the points on the curve is $7$. This means that any point on the curve generates it.

Example 6.2. A careful investigation of the elliptic curve $E:$ $y^2=x^3+2$ defined over the field $\mathbb{Z}_{7}$, yields that its order is $9$. Furthermore, it can be found that any point $P \in E$ yields $3P=\infty$. Therefore, the elliptic curve $E(\mathbb{Z}_7)$ defined by the above equation is isomorphic to the group $\mathbb{Z}_{3} \oplus \mathbb{Z}_{3}$.

From these two examples, it can be conjectured that the finite abelian group of points on any elliptic curve $E$ is either isomorphic to a cyclic group or the direct sum of cyclic groups.

Theorem 6.4. [3] Let $E$ be an elliptic curve defined over a finite field $\mathbb{K}$, then: $$E(\mathbb{K}) \simeq Z_{n} \hspace{1cm} or \hspace{1cm} \mathbb{Z}_{n_1} \oplus \mathbb{Z}_{n_2}$$ For some integer $n \geq 1$, or for some integers $n_1,n_2 \geq 1$, with $n_1 \mid n_2$.

Proof. The group of points on an elliptic curve is a finite abelian group. Any finite abelian group is isomorphic to the direct sum of cyclic groups: $$E(\mathbb{K}) \simeq \mathbb{Z}_{n_1} \oplus \mathbb{Z}_{n_2} \oplus ... \oplus \mathbb{Z}_{n_r}$$ with $n_i \mid n_{i+1}$ for $i \geq 1$. Therefore, each $\mathbb{Z}_{n_i}$ has $n_1$ elements of order dividing $n_1$, since $n_1\mid n_i$ for $1 \leq i$. So, the elliptic curve has $n^{r}_1$ points of order dividing $n_1$. But theorem 6.3 gurantees that there are at most $n^{2}_1$ such points. Therefore, $r$ can be at most $2$.

iii. Group order of Elliptic Curves

Given an elliptic curve $E$ defined over a finite field $\mathbb{K}_q$ (where $q$ is a power of a prime), what can be said about the number of points on $E$ ?

Theorem 6.4 (Hasse's theorem [6]) . Given an elliptic curve $E$ defined over the finite field $\mathbb{F}_q$, then the order of $E(\mathbb{F}_q)$ satisfies the following inequality: $$|\#E(\mathbb{F}_q)-(q+1)|\leq 2\sqrt{q}$$

Example 6.3. One can utilize Hasse's theorem to easily find the exact number of points on an elliptic curve without having to list them all down. Example 6.1 found that the point $(0,1)$ had order $9$. Assuming we don't know the order of the elliptic curve in Example 6.1, it will be denoted by $N=\#E(\mathbb{Z}_5)$. By Lagrange's theorem, the order of $(0,1)$ must divide $N$. Therefore, $N$ is a multiple of $9$. Using Hasse's theorem with $q=5$: $$5+1-2\sqrt{5} \leq N \leq 5+1+2\sqrt{5}$$ $$2 \leq N \leq 10$$ The only multiple of $9$ in this interval is $9$ itself. Therefore, $N=\#E(\mathbb{Z}_5)=9$ and the group is generated by the point $(0,1)$.

Example 6.4. [3] Let the elliptic curve $E$ be $y^2=x^3+7x+5$ over $\mathbb{Z}_{349}$. It can be verified that the point $(16,5)$ has order $191$. Therefore $N=\#E(\mathbb{Z}_{349})$ is a multiple of $191$. Using Hasse's theorem: $$313 \leq N \leq 387$$ The only value that's a multiple of $191$ in this range is $382$. Therefore, $\#E(\mathbb{Z}_{349})=382$. Using Hasse's theorem has cut the work down by half. Instead of trying to list down the $382$ points on the elliptic curve, one only needs to do have the calculations to get the order of $(16,5)$ and the order of $E(\mathbb{Z}_{349})$.

VII. Elliptic Curve cryptography

The reason to use elliptic curves in cryptographic situations is that elliptic curve cryptosystems can provide equivalent level of security of classical cryptosystems with fewer key sizes. This will reduce the chip size and reduce the power consumption for cryptosystems. In [8], RSA systems with key the following key sizes 1024/2048/3072-bit were replaced by equivalent elliptic curve cryptosystems with only 160/224/256-bit keys. Furthermore, key generation is faster in ECC than for RSA. In [14], experiments using an electronic device called 3Com’s PalmPilot, found that to generate a 512-bit RSA key, it took the device around 3.4 minutes. While for an equivalent 163-bit ECC-DSA key, it took around 0.597 seconds.

i. The Discrete Logarithm Problem

The security of RSA-based Cryptosystems depend on the computational in-feasibility of factoring large numbers, in a sense that it uses a one-way function to compute it easily but there's no efficient, non-quantum, polynomial time algorithm for factoring it.
Another class of cryptosystems depend on the computational in-feasibility of determining logarithms $\mod{n}$. This problem is called the Discrete Logarithm Problem (DLP).

Definition 7.1. (Discrete Logarithm Problem). Let $G$ be a finite cyclic group generated by $g$. Given $h \in G$, the DLP is to find the least positive integer $x$ such that: $$g^x = h$$ $x$ is called the discrete logarithm of $h$ with respect to $g$ and denoted by $x = \log_{g}{h}$.

In cyclic groups, one is always guaranteed to find such $x$. But if $G$ is not cyclic, then The DLP has solution if and only if $h \in \langle g \rangle$. If the order of $g$ is $n$, then $x \in \mathbb{Z}_{n}$, if it exists. Modular exponentiation serves as an example of a one-way function. A one-way function is a function for which given $x$, it's easy to compute $f(x)=y$ but it's computationally hard to inverse the operation. An example of a one-way function is multiplication of numbers: Given $a$ and $b$, it's easy to compute $n=ab$. However, there is no known polynomial time classical algorithm for factoring large integers. With modular exponentiation, given $g$ and $x$, it's easy to compute $g^x=h$. However, it's hard to get $x$ given $g$ and $h$ [7].

For additive groups (group of points on an elliptic curve in this case), the discrete logarithm problem can be stated as follows:

Definition 7.2. (DLP for additive groups). Let $G$ be a finite cyclic group generated by $P$. Given $Q \in G$, the DLP is to find the least positive integer $k$ such that: $$kP = Q$$

ii. Singular Curves

The reason that singular curves are not used in practise in cryptosystems is that there exists an isomorphism between the non-singular points on elliptic curve and another group, where the DLP for elliptic curves become easier to solve. Therefore, the elliptic curves advised for use in cryptographic purposes are non-singular [7] [3].

Example 7.1. In Section 5.3, it was proven that there exist an isomorphism between the non-singular points on the elliptic curve $y^2=x^3$ defined over the field $\mathbb{K}$ and the additive group $K$. Let's define the finite field $\mathbb{K}$ to be $\mathbb{Z}_5$. Consider the multiples of the point $P=(1,1)$: $$P=(1,1), \hspace{0.2cm} 2P=(4,2), \hspace{0.2cm} 3P=(4,3), \hspace{0.2cm} 4P=(1,4)$$ As demonstrated, each multiple of $P=(1,1)$ can be represented in the following form:$$mP=(m^{-2},m^{-3})$$ Where each coordinate is calculated mod $5$. It can be noticed that each point corresponds to an integer $a = x/y$ as was demonstrated in Section 5.3. The corresponding integers are $1,2,3,4$, respectively. Therefore, from the isomorphism mentioned, addition of non-singular points on the elliptic curve correspond to addition of integers in the additive group $\mathbb{Z}_{5}$. Now, given $P$ and $mP$, the isomorphism helps to get the integer $m$ and attack the DLP for elliptic curves easily.

iii. Attacks on The discrete Logarithm Problem

One can try to attack the discrete logarithm problem for elliptic curves by brute forcing all the possible values for $m$. However, this method isn't computationally efficient when $m$ can be an integer with hundreds of digits. One attack on the general discrete logarithm problem is by using index calculus. However, it's not suitable, generally, for all groups and applies to the multiplicative group of a finite field. Other attacks that are applicable for elliptic curves that will be discussed are Baby Step, Giant Step, Pollard's $\rho$-method, and The Pohlig-Hellman Method [9] . Throughout this section, we will use the additive groups discrete logarithm problem since we are interested in elliptic curves. Define an elliptic curve $E$ over the finite field $\mathbb{K}$. Furthermore, let $P$ and $Q$ be two points $\in E(\mathbb{K})$ such that $kP=Q$ for some integer $k$. Let the order of the group generated by $P$ be $N$.

iii.i. Baby Step, Giant Step [12]

Suppose Eve wants to solve the DLP $Q=kP$, then they can use the following algorithm. The following algorithm has approximately $\sqrt{N}$ steps and uses up approximately $\sqrt{N}$ storage. So, it can be only used with points $P$ with moderate order $N$.

	Algorithm 1: Baby Step, Giant Step
1	Choose an integer $m$ with $m^2\geq N$.
2	Make a list with each of the points $iP$ for $0\leq i < m $.
3	Make another list with the points $Q-jmP$ for $0\leq j < m$ and stop computing until an point matches another point in the first list.
4	If there's a match for some $i$ and $j$ such sthat $iP=Q-jmP$, then if $Q=kP$, $k\equiv i+jm \mod{N}$

Why should there exist a match between the two lists ? Since one can write $0 \leq k < N \leq m^2$, $k$ can be written in base $m$: $$k=k_{0}+k_{1}m$$ where $0 \leq k_0,k_1 < m$. If we let $k_0=i$ and $k_1=j$, then a match exists between the two lists: $$Q-k_1mP=kP-k_1mP=k_0P$$

iii.ii. Pollard's $\rho$-method

Pollard's $\rho$-method is currently the best-known algorithm used for attacking the DLP for EC. However, its time complexity is full exponential $\mathcal{O}(\sqrt{n})$. Even though both Pollard's $\rho$-method and the Baby Step, Giant Step attack have the same running time, Pollard's $\rho$-method takes up only little storage. This because Pollard's $\rho$ method store only the current pair of points instead of just storing all the points until a match collision occurs. The basic idea of Pollard's $\rho$ method is random walks over a finite group $G$. A choice of a function $f$ that maps $G$ into itself ($f:G \rightarrow G$) is made such that the function behaves "randomly". Then a random start "point" $P_0 \in G$ is chosen. Then, we perform the following iterations until we get a match collision: $$P_{i+1}=f(P_i)$$ Notice that a collision must occur since $G$ is finite. Let $i_0$ and $j_0$ be the smallest indices such that $P_{i_0} = P_{j_0}$. Then, for any $l$, we have: $$P_{i_0+l} = P_{j_0+l}$$

These iterations generate a shape similar to the Greek letter $\rho$, as shown in figure 5. The values of $P_i$ up to $i=5$ is called the tail. Once the iterations reach $P_5$, the function iterates through a closed loop, with a period $l$. How is the random-walk function $f$ is chosen? How is the starting point $P_0$ of the random walk is randomly chosen? How can detecting a collision match be used to solve the DLP for EC? This is answered by the algorithm on the next page.

	Algorithm 2: Pollard's $\rho$ method
1	Divide an additive group $G$ with order $N$ into $s$ disjoint sets with approximately the same size: $G = \bigcup_{i=1}^{s} S_i$
2	For each $S_i$ define a step $M_i$ by randomly choosing two integers $a_i,b_i \mod N$: $M_i=a_iP+b_iQ$
3	Define the function $f:G \rightarrow G$ as follows: $$f(g) = g + M_i \hspace{0.5cm} \text{if } g \in S_i$$
4	Randomly choose two integers $a_0,b_0 \mod{N}$ and let the point $P_0=a_0P+b_0Q$ be the starting point of the random-walk.
5	Perform the iterations until a match is found for some $j_0>i_0$: $$P_{j_0} = P_{i_0} \rightarrow u_{j_0}P+v_{j_0}Q=u_{i_0}P+v_{i_0}Q$$
6	Let $d=gcd(v_{j_0}-v_{i_0}, N)$, then: $$k \equiv (v_{j_0}-v_{i_0})^{-1}(u_{i_0}-u_{j_0}) \mod N/d$$

How does this yield the integer $k$? First notice that the statement $P_{j_0} = P_{i_0}$ can be stated in another way: $$u_{j_0}P+v_{j_0}Q=u_{i_0}P+v_{i_0}Q \rightarrow (u_{i_0}-u_{j_0})P = (v_{j_0}-v_{i_0})Q$$ So, if we can find $(v_{j_0}-v_{i_0})^{-1} \mod N/d$, then we can multiply both sides by this integer to get: $$(v_{j_0}-v_{i_0})^{-1}(u_{i_0}-u_{j_0})P = Q$$ Therefore, we have found the integer $k$ such that $kP=Q$.

iii.iii. The Pohlig-Hellman Method [11]

Let $P,Q \in E(\mathbb{K})$ such that $Q=kP$. Let $N$ be the order of the point $P$ with prime factorization as follows: $$\prod_{i}p^{r_i}_{i}$$ The idea of the Pohlig-Hellman Method is to compute $k \mod{p^{r_i}_{i}}$ for each $i$. If this can be done, the resulting simultaneous congruences can be combined using the Chinese remainder theorem to find $k$. Let $q$ be one of the primes dividing $N$, and let $r$ be the largest integer such that $q^r \mid N$. Then, the integer $k$ can be written in the base expansion of $q$ as: $$k=k_0+k_1q+k_2q^2+...$$

	Algorithm 3: The Pohlig-Hellman Method
1	Make a list $S$ with the following points as elements $S=\{j \left( \frac{N}{q}P\right)\mid 0 \leq j \leq q-1$
2	Find the point $\frac{N}{q}Q$, which will be the element $k_0 \left( \frac{N}{q}P \right)$ of S.
3	If $r=1$, then $k \equiv k_0 \mod{q}$. Otherwise, continue the steps
4	Find the point $Q_1=Q-k_0P$, and the point $\frac{N}{q^2}Q_1$, which will be another element $k_1 \left( \frac{N}{q}P \right)$ of S.
5	If $r=2$, then $k \equiv k_0+k_1q \mod{q^2}$.
6	In general, $Q_i=Q_{i-1}-k_{i-1}q^{i-1}P$
7	Find the point $\frac{N}{q^{i+1}}Q_i$, which will be the element $k_i \left( \frac{N}{q}P \right)$ of S.
8	If $i=r-1$, then $k \equiv k_0+k_1q+...+k_{r-1}q^{r-1} \mod{q^r}$.

Why should the point $\frac{N}{q^{i+1}}Q_i$ be an element of the set $S$? Consider the following: \begin{align} \frac{N}{q}Q =\frac{N}{q}kP &= \frac{N}{q}(k_0+k_1q+...)P \\ &=k_0\frac{N}{q}P+(k_1+k_2q+...)NP\\ &=k_0\frac{N}{q}P + \infty\\ &=k_0\frac{N}{q}P \end{align}

Similar calculations will show that every point $\frac{N}{q^{i+1}}Q_i \in S$. The algorithm must stop when $i=r-1$ for two reasons: First, if the algorithm continues and $i=r$, $\frac{N}{q^{r+1}}$ won't be an integer and we don't even have to get further values of $k_i$ since we had already attained $k \mod{q^r}$.

iv. Elliptic Curves Diffie-Hellman Key Exchange (ECDH) [3]

Suppose two parties $X$ and $Y$ want to communicate over an elliptic curve cryptosystem. However, they haven't met in advance to determine the key for communication. It also might be expensive to trust a courier to transfer the key between $X$ and $Y$. A method introduced by Diffie and Hellman \cite{Diffie} helps establishing a common secret key between the two parties.

	Algorithm 4: Elliptic Curves Diffie-Hellman Key Exchange (ECDH)
1	$X$ and $Y$ choose and elliptic curve $E$ defined over the finite field $\mathbb{K}_{q}$. The curve $E$ is chosen such that the discrete logarithm problem for elliptic curves is computationally infeasible to solve. Furthermore, they choose a point $P \in E(\mathbb{K}_{q})$ such that the cyclic subgroup $\langle P \rangle$ has a large prime order.
2	$X$ chooses a random integer $m$ and computes $P_m=mP$. Then, the message $P_m$ is sent to the party $Y$, while the integer $m$ is kept secret.
3	$Y$ does the same process, but it chooses another random secret integer $n$. $Y$ compute $P_n=nP$ and sends it to $X$.
4	$X$ computes $mP_n=mnP$.
5	$Y$ computes $nP_m=mnP$.
6	After that, the two parties agree on a method to extract the secret shared key from $mnP$. A hash function on the $x$ or $y-coordinates$ of $mnP$ can be used to extract the secret key.

The information made public in all the communications done by the two parties to establish the secret key is the elliptic curve $E(\mathbb{K}_{q})$, the point $P$, and the quantities $P_m=mP$ and $P_n=nP$. While the integers $m$ and $n$ are kept secret. If a third party wants to know the secret key, it would have to solve the Diffie-Hellman problem, which is basically DLP for elliptic curves.

Definition 7.3 (Diffie-Hellman Problem). Given $P$, $nP$, and $mP \in E(\mathbb{K}_{q})$, compute $mnP$.

In order to compute $mnP$ from $nP$ or $mP$, the third party would at first has to solve for either $m$ or $n$. This defines a discrete logarithm problem. If the elliptic curve is chosen wisely, the process of determining the shared secret $mnP$ between the two parties will be computationally infeasible.

v. Representing Messages as Points on Elliptic Curves

Before discussing the encryption and signature schemes in the following sections, we present a way of representing messages $m$ as the $x-coordinates$ of points that was proposed by Kobiltz [3]. Since no deterministic algorithm with polynomial time for reprsenting messages as points is known, the method proposed by Kobiltz is probabilistic. A naive representation of the message $m$ is to use it as the $x-coordinate$ for a point on the elliptic curve: $$y^2=m^3+Ax+B$$ However, the probability that this equation doesn't have a solution $\in \mathbb{F}_{p}$ is $1/2$. Therefore, the message $m$ is modified by adding to it a few bits. First, large integer $N$ is chosen such that a failure rate of being able to find a $y$ of about $1/2^N$ would be acceptable. Then, Let the message be expressed as a number $m$, such that: $$0 \leq m < \frac{p}{N}$$ Then, use $m$ and $N$ to calculate the following number: $$x=mN+j \hspace{1cm} \text{for } 0 \leq j < N$$ After that, we try to find the solutions to the equation $y^2=x^3+Ax+B$ by plugging in every possible value of $x$ as $j$ ranges over all the values from $0$ to $N-1$. We perform this process until an integer $j$ has been found such that $x^3+Ax+B$ has a square root $\in \mathbb{F}_{p}$. Since for every value of $j$ there's a $1/2$ of failure, then this method overall has a probability of failure of around $1/2^N$. To retrieve a message $m$ from the $x-coordinate$, we simply calculate: $$m=\left\lfloor \frac{x}{N} \right\rfloor$$

vi. Massey-Omura Encryption

The Massey-Omura Encryption requires the two parties to do many communications for sending only one message. This why it's rarely used in practise. However, investigating this encryption scheme still gives us some insight on the general mechanism of elliptic curve cryptosystems.

	Algorithm 5: Massey-Omura Encryption
1	$X$ and $Y$ choose and elliptic curve $E$ defined over the finite field $\mathbb{K}_{q}$. The curve $E$ is chosen such that the discrete logarithm problem for elliptic curves is computationally infeasible to solve. Let the order of the elliptic curve be $N=\#E(\mathbb{K}_q)$
2	$X$ wants to send the message $M$ to $Y$ by first representing it as a point $P \in \mathbb{K}_{q}$
3	$X$ chooses a random secret integer $a$ with $gcd(a,N)=1$ (to be able to find its inverse mod $N$), and computes $M_1=aM$. $X$ sends $M_1$ to $Y$
4	$Y$ does the same procedure and chooses a random secret integer $b$ with $gcd(b,N)=1$, computes the point $M_2=bM_1=abM$ and sends it to $X$
5	Now, $X$ computes the point $M_3=a^{-1}M_2$, where $a^{-1} \in \mathbb{Z}_{n}$ and sends it back to $Y$.
6	$Y$ computes the point $M_4=b^{-1}M_3$, where $b^{-1}\in \mathbb{Z}_{N}$. $M_4=M$, the message which $X$ wanted to send to $Y$.

There's only one thing to be noted. From the final point calculated by $Y$, we have: $$M_4=b^{-1}a^{-1}baM=M$$ If we are working $\mod{N}$, there would be no problem. But how can we justify that $a$, for example, cancels $a^{-1} \mod N$ in $M_4$ if we are not working $\mod N$?

Theorem 7.1. Define an elliptic curve $E$ over the field $\mathbb{K}$, with $N=\#E(\mathbb{K})$. Let $P \in E(\mathbb{K})$. Given a random integer $a$, with $gcd(a,N)=1$, and $a^{-1} \mod N$, then: $$aa^{-1}P=P$$

Proof. We have $aa^{-1} \equiv 1 \mod{N}$. Therefore, $aa^{-1}=qN+1$ for some integer $q$. Since the order of the group $E$ is $N$, we have by Lagrange theorem $NQ=\infty$ for any point $Q \in E(\mathbb{K})$. Therefore: $$aa^{-1}P = (qN+1)P = qNP+P=q\infty + P = P$$

It can be shown that this scheme results in a Diffie-Hellman Problem. If we let $P=abM$, $m=a^{-1}$, and $n=b^{-1}$, then the public information are $P=abM$, $aM=b^{-1}P=nP$, and $bM=a^{-1}P=mP$. To get $M$, Eve would have to compute $M = a^{-1}b^{-1}P=mnP$, given $P$, $nP$, and $mP$.

vii. ElGamal Public Key Encryption

Suppose $X$ wants to send a message $M$ to $Y$ over an ECC based on ElGamal Public Key Encryption. He establishes his public key as follows: Like always, $Y$ chooses an elliptic curve $E$ defined over a finite field $\mathbb{K}$ such that the DLP is computationally infeasible in $E(\mathbb{K})$. $Y$ chooses a point $P \in E(\mathbb{K})$. Then, he picks a random secret integer $a$ and computes the point $Q=aP$. Then, $Y$ makes public the elliptic curve $E(\mathbb{K})$, the points $P$ and $Q$. Collectively, this information is $Y's$ public key. $X$ sends a message $Y$ to $X$ using $Y's$ public key as follows:

	Algorithm 6: ElGamal Public Key Encryption
1	Using the elliptic curve made public by $Y$, $X$ represents the message as a point $M \in E(\mathbb{K})$.
2	$X$ randomly generates a secret integer $b$ and computes the points $M_1=bP$ and $M_2=M+bQ$.
3	$X$ sends both $M_1$ and $M_2$ to $Y$.
4	$Y$ decrypts the message as follows: $$M=M_2-aM_1$$

To verify that the point $M_2-aM_1$ is the message $M$: $$M_2-aM_1=M+bQ-abP=M+abP-abP=M$$ It's crucial to use a different $b$ for each message. Otherwise, Eve would be able to deduce any message given a known message. Suppose $X$ uses the same $b$ for two different messages $M$ and $M^\prime$. Then Eve knows that $X$ used the same $b$ since $M_1=M^{\prime}_1=bP$. Now, given the known message $M$, Eve can deduce the message $M^\prime$ as follows: $$M^\prime = M - M_2 - M^{\prime}_2$$ Again, this Encryption scheme also defines a Diffie-Hellman Problem: Given $P$, $Q=aP$, and $M_1=bP$, compute $aM_1=abP$.

viii. ElGamal Digital Signatures

Suppose that the communicating party $X$ wants to sign a document and send it to $Y$ electronically. Instead of appending $X's$ signature to the document, $X$ performs a number of transformations on the document to obtain the signature in such a way that the signature can't be forged and used again. First, $X$ establishes with $Y$ an elliptic curve $E(\mathbb{K})$. In addition, $X$ chooses a point $P \in E(\mathbb{K})$ with order $N$ and a random secret integer $a$ and computes the point $B=aP$. Furthermore, $X$ chooses a function as follows: $$\psi: E(\mathbb{K}) \rightarrow \mathbb{Z}$$ The information made public by $X$ are $E(\mathbb{K})$, $\psi$,$P$ and $B$. The only piece of information kept secret is the random secret integer $a$. To sign a document, $X$ does as follows:

	Algorithm 7: ElGamal Digital Signature
1	Represent the message as an integer $m < N$.
2	Picks a random integer $b$ with $gcd(b,N)=1$ and computes the point $R=bP$.
3	Then, $X$ calculates the signature $s \equiv b^{-1}(m-a\psi(R))$.
4	$X$ sends the signed message $(m,R,s)$ to $Y$.

For $Y$ to verify the signature of $X$ on the document, it uses $X's$ public information to calculate the following quantities: $$Y_1=\psi(R)B+sR \hspace{1cm} and \hspace{1cm} Y_2=mP$$ If $Y$ finds that $Y_1=Y_2$, then they declare the signature of $X$ on the document valid. That's because if the signature is valid, then: $$ Y_1 = \psi(R)B + sR = \psi(R)aP + skP = \psi(R)aP + (m - a\psi(R))P = mP = Y_2$$ If Eve wishes to forge $X's$ signature, they would have to solve for $a$ given $P$ and $B=aP$, which is just a discrete logarithm problem for elliptic curves.

ix. Performance Comparison of EC and RSA Cryptosystems

The main advantage that EC cryptosystems offer over the more popular RSA public-key cryptosystems is their ability to provide an equivalent level of security with much smaller key sizes. EC cryptosystems are getting popular and used in leading companies, where it can also be used with RSA. For example, some companies as Amazon and Linkedin use ECDH RSA protocols (Elliptic Curve Diffie-Hellman Key Exchange with RSA) [8]. In addition, study [17] has proposed an E-voting system where each vote is represented as a point on the elliptic curve. The secrecy of each vote is achieved by the hardness of the DLP for elliptic curves contained in the ElGamal Encryption scheme. The E-Voting system achieved privacy, universal verifiability and robustness of e-voting. Furthermore, it demonstrated that it acquired the same level of security that an e-voting system based on RSA but with much smaller key size. Another appealing aspect of EC cryptosystems is that the time complexity for the best-known algorithm, Pollard's $\rho$ algorithm, to solve the DLP for EC is full exponential $\mathcal{O}(\sqrt{n})$ [13]. On the other hand, the time complexity of the best-known algorithm for factoring integers, Pollard's version of general number field sieve, for RSA is sub-exponential. In study [16] , a comparative analysis was conducted on the encryption schemes of both RSA and EC. The encryption schemes that were used in the EC crypotsystem were ElGamal Elliptic Curve Encryption algorithm and Menezes-Vanstone Elliptic Curve Encryption algorithm. The study implemented the encryption schemes using the FlexiProvider library in java. The data collected were on the time required for using the key generation, encryption, and decryption algorithms. It concluded that the EC cryptosystems can provide a level of security that's equivalent to other RSA cryptosystems with larger keys. However, it pointed out that EC cryptosystems still need to be studied extensively to be able to replace RSA cryptosystems. In another study [8] , an investigation for the efficiency of RSA with its two variants, Basic RSA and RSA with Chinese Remainder Theorem, and ECC was done. As shown in figure 6, the study concluded that the ratio between the key sizes of both variants of RSA to the key size of an equivalent AES security level ECC was growing exponentially. The study also examined the encryption and decryption time for 27-bit data with RSA and ECC with equivalent AES security levels of 128-bit. Even though the encryption time for ECC (0.2604 secs) was found to be higher than the two variants of RSA (0.1563 secs), the decryption time was much significantly lower, around 0.1042 secs for ECC and 0.8906 secs for RSA, as shown in figures 7 and 8. This accounts for an overall total time of encryption and decryption of about 0.3646 secs for ECC and 1.151 secs for RSA, as shown in figure 9.

VIII. Isogeny-Based Cryptography

Even though ECC proves to be resistant and efficient against attacks by classical algorithms, it will not be able to stand the threat posed by post-quantum attacks. In particular, it's known that the DLP for elliptic curves can be attacked by Shor's Algorithm in polynomial time when the algorithm is run on a quantum computer [19] . The reason for that is because the DLP for elliptic curves is defined over abelian groups. Therefore, quantum resistant schemes for DLP of elliptic curves will heavily depend on non-abelian groups [20] , which Shor's Algorithm can't attack.

i. Isogenies

The shared secret key established by communicating parties in the classical Elliptic Curve Diffie-Hellman key exchange depends on the scalar multiplication. However, a post-quantum attack will be easily able to break cryptosystems based on scalar multiplication. Therefore, a quantum safe procedure was proposed, where elliptic curves are mapped to another elliptic curves through rational maps called isogenies.

Definition 8.1 (Rational Map [3]). A map $\phi: E_0 \rightarrow E_1$ between two elliptic curves is called a rational map if it maps points to their images whose coordinates are represented as a ratio of polynomials $$\phi(P)=\phi(x,y)=\left(\frac{p_1(x,y)}{q_1(x,y)},\frac{p_2(x,y)}{q_2(x,y)}\right) \hspace{1cm} \text{for any point } P \in E_0$$

If the rational map is a homomorphism, it respects addition $(\phi(P+Q) = \phi(P) + \phi(Q))$, then it's called an isogeny.

Definition 8.2 (Isogeny [3] ). An isogeny $\phi$ is a rational map between two elliptic curves $\phi: E_0 \rightarrow E_1$ that is a group homomorphism. Every isogeny maps $E_0$ onto $E_1$ (surjective).

Definition 8.3 (Kernel [10] ). The kernel of an isogeny $\phi : E_0 \rightarrow E_1$, denoted $ker \hspace{0.1cm} \phi$, is the set of points that map to the identity of $E_1$, namely, $$ker \hspace{0.1cm} \phi = \{P \mid P \in E_0 \text{ and } \phi(P) = \infty\}$$ Intuitively, we can think of the kernel as the inverse image of $\infty$

The classical ECDH key exchange, as presented in section 7.4, allow the two communicating parties to reach the same shared key $mnP$. In a quantum-safe key exchange, however, we would like to replace the scalar $m$ by two the isogenies $\phi_m$ and $\psi_m$. We would also like to replace $n$ by the two isogenies $\phi_n$ and $\psi_m$. In addition, a key difference between the two schemes will be the private and public keys. In ECDH, $X's$ public key is the point $mP$, an image of the point $P$ under scalar multiplication mapping. However, in a quantum-safe scheme, the public keys will be whole image curves under isogenies and not points. Another distinction is that in ECDH, the two communicating parties eventually reach exactly the same point $mnP$. However, the isogenies yield different image elliptic curves $E_{mn}$ and $E_{nm}$. Although the curves are different, their group structure are identical and the two curves are isomorphic $E_{mn} \simeq E_{nm}$. We present a quantity called the $j-invariant$ for elliptic curves, which is helpful to determine isomorphism.

Definition 8.4 ($j-invariant$ [3]). The $j-invariant$ of the elliptic curve $E: y^2=x^3+Ax+B$ is the following quantity: $$j(E) = \frac{6912A^3}{4A^3+27B^2}$$

Proposition 8.1 ([3] ). Two elliptic curves $E_0$ and $E_1$ are isomorphic, if and only if, $$j(E_0)=j(E_1)$$

ii. Generating the Private/Public key pair

The private key of a quantum-safe scheme will be isogenies. Just like the integers $m$ and $n$, isogenies are generated randomly. First, the kernel of an isogeny is generated through a random point that is selected from the elliptic curve. Before we discuss the random generation of isogenies and the private/public key generation, we have to first describe the finite fields over which elliptic curves are defined over in isogeny-based cryptography.

ii.i. Elliptic Curves over $\mathbb{F}_{p^2}$

In classical elliptic curve cryptography we generally use the field $\mathbb{F}_{p}$. However, quantum-safe isogeny-based cryptography uses supersingular elliptic curves defined over $\mathbb{F}_{p^2}$ [21] . The prime $p$ will always be used such that $p \equiv 3 \mod{4}$ and $p=2^{a}3^{b}-1$. In general, elements of $\mathbb{F}_{p^2}$ have the following form: $$\mathbb{F}_{p^2}= \{a+b\alpha| \alpha^2 \equiv -1, a,b \in \mathbb{F}_{p}\}$$ of all the points on $E(\mathbb{F}_{p^2})$, we are particularly interested in the points of the torsion groups $E[2^a]$ and $E[3^b]$ for two reasons. First of all, when generating isogenies, it's important to have their kernels be a power of a small prime to calculate isogenies more easily. Therefore, the most suitable points for the kernels of the isogenies would have to belong to either of the two torsion groups: $E[2^a]$ or $E[3^b]$. Secondly, the group structure of $E[2^a]$ and $E[3^b]$ has a nice property as demonstrated in the next theorem.

Theorem 8.1 ([20] ). There exists points $P_m,Q_m,P_n,P_m \in E(\mathbb{F}_{p^2})$ such that: $$E[2^a]=\langle P_m, Q_m \rangle$$ $$E[3^b]=\langle P_n, Q_n \rangle$$

iii. Isogeny Generation

As in the classical ECDH key exchange scheme, each communicating party $X$ and $Y$ generate both a private key and a public key. The private key of $X$ is the cyclic isogeny they randomly generate. A random point $R_m$ is chosen on the curve and the subgroup generated by the point $R_m$ will be the kernel for the isogeny. The isogeny is created from the kernel using Vélu’s Formulas. Therefore, the isogeny random point $R_m$ and it's associated isogeny $\phi_m$ will be $X's$ private key.

How is a random point $R_m$ chosen? since $R_m \in E[2^a]$, we know from theorem 8.1 that any point $N \in E[2^a]$ can be written in the following form: $$N = xP_m+yQ_m \hspace{0.7cm} (x,y \in \mathbb{F}_{p^2})$$ Therefore, for $X$ to generate a random point point $R_m \in E[2^a]$, a random seed $r_m$ is chosen with: $$0 \leq r_m < 2^a$$ Using this seed, $X$ constructs the point $R_m=P_m+r_mQ_m$, which belongs to the torsion group $E[2^a]$. $X$ then generates the cyclic group $\langle R_m \rangle$, which will be the kernel of the isogeny $\phi_m$. Using Vélu’s Formulas, $X$ constructs an isogeny $\phi_m$ out of its kernel $\langle R_m \rangle$. Similarly, $Y$ construct their private key by choosing a random seed $r_n$ with: $$0 \leq r_n < 3^b$$ After that, they generate the point $R_n=P_n+r_nQ_n$ that belongs to the torsion group $E[3^b]$ and generates the cyclic isogeny $\phi_n$ from its kernel $\langle R_n \rangle$ in the same manner using Vélu’s Formulas.

After that, they construct another two isogenies $\psi_m$ and $\psi_n$. In order for $X$ to construct $\psi_m$, they need to have $\phi_n(P_m)$ and $\phi_n(Q_n)$. So, $Y$ includes both image points, $\phi_n(P_m)$ and $\phi_n(Q_n)$, in their public key accompanied with the image curve $E_n = \phi_n(E)$ After knowing them, $X$ constructs the following point: $$\phi_n(R_m)= \phi_n(P_m) + r_m \phi_n(Q_m)$$ Then, $X$ constructs the isogeny $\psi_m$ with $ker \hspace{0.1 cm} \psi_m = \langle \phi_n(R_m) \rangle$. Similarly, $X$ will include $E_m = \phi_m(E), \phi_m(P_n), \phi_m(Q_n)$ in their public key. Then, $Y$ uses this information to construct the cyclic isogeny $\psi_n$ with $ker \hspace{0.1cm} \psi_n = \langle \phi_n(R_n) \rangle$. After getting each of the isogenies $\phi_m, \phi_n, \psi_m, \psi_n$, both $X$ and $Y$ apply their isogenies on the elliptic curve in order to reach a shared-secret.

Lemma 8.1 ([3] ). If two isogenies have the same kernel, then the image curves are isomorphic.

Theorem 8.2 ([20] ). Let $\psi_m$ be the isogeny with $ker \hspace{0.1cm} \psi_m =\langle \phi_n(R_m) \rangle$, and let $\psi_n$ be the isogeny with $ker \hspace{0.1cm} \psi_n =\langle \phi_m(R_n) \rangle$ . Then, $$ker \hspace{0.1cm} \psi_m \circ \phi_n = ker \hspace{0.1cm} \psi_n \circ \phi_m$$ Therefore, $$E_{mn} = \psi_m \circ \phi_n(E) \simeq \psi_n \circ \phi_m(E) = E_{nm}$$

Even though both communicating parties don't reach the same mirror elliptic curve, they achieve two curves whose isogenies' kernels are identical. Therefore, by lemma 8.1, the image curves $E_{mn}$ and $E_{nm}$ are isomorphic. Consequentially, by proposition 8.1, the shared secret between $X$ and $Y$ is the $j-invariant$.

iv. Supersingular Isogeny Diffie–Hellman key exchange (SIDH) [18]

Algorithm 8: Supersingular Isogeny Diffie–Hellman key exchange (SIDH)

1

Public Parameters: Supersingular elliptic curve $E: y^2=x^3+ax+b$ defined over $\mathbb{F}_{p^2}$ for $p = 2^a3^b-1$. Generator points $P_m, Q_m$ and $P_n, Q_n$ such that $E[2^a]=\langle P_m, Q_m \rangle$ and $E[3^b] = \langle P_n, Q_n \rangle$. Key Generation: $X$ randomly chooses an integer $r_m$ with $0 \leq r_m < 2^a$. $X$ generates the random points $R_m \in E[2^a]$ given by $R_m = P_m+r_mQ_m$. $X$ calculates the isogeny $\phi_m$ with the following kernel: $ker \hspace{0.1cm} \phi_m = \langle R_m \rangle$ using Vélu’s Formulas. $X's$ private key is the isogeny $\phi_m$ $X$ calculates the image curve $E_m = \phi_m(E)$ and sends her public key $(E_m,\phi_m(P_n),\phi_m(Q_n)$ to $Y$. $Y$ uses $X's$ public key to calculate the isogeny $\psi_n$. $Y$ does the same procedure with a random integer $0\leq r_n < 3^b$, calculates the kernel generated by $R_n=P_n + r_nQ_n$ and its associated isogeny $\phi_n$, and sends its public key $(E_n,\phi_n(P_m),\phi_n(Q_m)$ to $X$ to calculate the isogeny $\psi_n$. Shared Secret $X$ calculates the image curve $E_{mn} = \psi_m \circ \phi_n(E)$ with $ker \hspace{0.1cm} E_{mn} = \langle R_m, R_n \rangle$. $Y$ calculates the image curve $E_{nm} = \psi_n \circ \phi_m(E)$ with $ker \hspace{0.1cm} E_{nm} = \langle R_m, R_n \rangle$. $X$ and $Y$ can both use the $j-invariant$ as their shared secret.

The previous algorithim provides a comprehensive guide for the process of key exchange for a quantum-resistant scheme. Figure 10 demonstrates a visual representation for the algorithm. For an adversary to be able to attack this scheme, they will have to find the kernel of an isogeny (and thus the isogeny itself) given an elliptic curve $E$ and its image curve $E^\prime$.

Definition 8.5 (The $l^e-isogeny$ problem). Given an elliptic curve $E$ and its image curve $E^\prime = \phi(E)$ over an isogeny $\phi$ with a kernel of size $l^e$, the $l^e-isogeny$ problem is to find $ker \hspace{0.1cm} \phi$ given only $E$ and $E^\prime$.

To ensure the security of SIDH scheme for $X$, the $2^a-isogeny$ problem must be computationally hard to ensure the security of $X's$ private key. Similarly, the $3^b-isogeny$ problem for $Y$ must also be computationally hard. Unlike ECDH, it's believed that SIDH can resist quantum attacks algorithms such as Shor's algorithm. However, it's a newly proposed scheme that still requires a lot of research.

IX. Conclusion

In this paper, we have covered the basic principles that govern the arithmetic of elliptic curves. An investigation of the torsion groups of points on an elliptic curve has resulted in identifying them as being isomorphic to the direct sum of two identical cyclic groups (or two cyclic groups with the order of one dividing the other). This result has been used to prove that points on a non-singular elliptic curve form a group that's isomorphic to a cyclic group or the direct sum of two cyclic groups. Furthermore, we examined the isomorphisms that arise between non-singular points on a singular elliptic curve and other groups. It has been shown that these isomorphisms make solving the DLP for EC easier on singular curves. Therefore, they shouldn't be used in cryptography. It has also been found that encryption schemes for EC, such as ElGamal Public Key Encryption, provide an equivalent level of security of another RSA cryptosystem but EC uses significantly lower key size. In addition, EC cryptosystems demonstrated a much lower time lapse of key generation, encryption, and decryption of data than RSA. This renders EC cryptosystems superior to RSA. However, due to EC cryptography not being extensively studied as RSA is, RSA remains to be the dominating cryptosystem to this day. However, the popularity of EC cryptosystems is increasing and is expected to hold a much stronger ground against attacks by quantum algorithms in the future using Isogeny-Based Cryptography.

I. Introduction

The concept of “War” in human Culture have been evolved through numerous eras. War can be started as a result of a strong competition between different human groups, which leads to many activities caused by aggressiveness and material huge consumption. The goal is to conquer other lands and take off its goods. Today, the decision to make war comes from political leaders who are often guided by their own interests and aggressions. Modern countries appear to be intrinsic to the character of humans, along with the intelligent practical skills to devise new tactics and technologies for warfare [3] .

As shown above, some reasons push us to live on another planet. So, there is some investigation into the possibility of life on other planets. The exoplanet was located by "The Kepler space telescope" on July 23, 2015, and "NASA" announced its discovery. The planet is located 550 pc (or 1,800 light-years) from the Solar System [4] . The name of this exoplanet was "Kepler-452b". It is the only planet in the system that Kepler has found, and it is a super-Earth exoplanet that orbits within the inner edge of the habitable zone of the sun-like star Kepler-452. Additionally, Kepler-452b likely has a thick atmosphere and liquid water, according to "NASA".

The answer depends on whether a planet is in the habitable zone, which is frequently defined as the constrained range of distances between a planet and its parent star that would permit liquid water to exist on its surface [5] . And according to "NASA," Traditional models of what is habitable assume a small, rocky, and water-rich planet similar to our own. And to simulate the climate, The Community Climate System Model, version 3, is the model used here to simulate life (CCSM3). It is a fully coupled atmosphere-ocean general circulation model in three dimensions (AOGCM). The model has been modified to study the climates of exoplanets after being created to study Earth’s climates [6].

One of the most famous scientific contributions to extraterrestrial life on other planets – yet, it is not widely accepted by the academic community – is the famous equation of the astronomer Frank Drake, named after his name. The Drake equation is composed of:

$$N = R * f_p * n_e * f_l * f_i * f_c * L$$

(1)

Where:
$N$ = number of civilizations with which humans could communicate
$R$ = mean rate of star formation
$f_p$ = fraction of stars that have planets
$n_e$ = mean number of planets that could support life per star with planets
$f_l$ = fraction of life-supporting planets that develop life
$f_i$ = fraction of planets with life where life develops intelligence
$f_c$ = fraction of intelligent civilizations that develop communication
$L$ = mean length of time that civilizations can communicate

Although it lacks scientific bases as it has been made depending on logical statements and some observations made [7] , there are about 100 scientific research questions in progress to find solutions for each factor of this equation, all constructed under the supervision of SETI Institution. This equation made a significant step towards the search for another life on other planets, despite the fact it hasn’t been solved completely to this day. So, could life exist on "another planet"?

The answer is dependent on whether a planet is in the habitable zone, which is frequently defined as the constrained range of distances between a planet and its parent star that would permit liquid water to exist on the planet's surface [8] . And according to "NASA," Traditional models of what is habitable assume a small, rocky, and water-rich planet similar to our own. And to simulate the climate, The Community Climate System Model, version 3, is the model used here to simulate life (CCSM3). It is a fully coupled atmosphere-ocean general circulation model in three dimensions (AOGCM). The model has been modified to study the climates of exoplanets after being originally created to study Earth’s climates [9].

II. Methods and results

i. Water

First and foremost, to call a planet a habitable area. We should identify the "habitable area." The area where a rocky planet can sustain liquid water on its surface is the habitable zone (HZ) around a star. That definition is appropriate because it allows for the possibility that the planet harbors an abundance of carbon-based, photosynthetic life that could alter the atmosphere in a way that could be remotely detected. However, the specific requirements for maintaining liquid water are still debatable [10].

Moreover, to produce livable conditions on mars, as hydrogen production from the water rises, the coverage area of the investigated territory can be expanded by increasing the number of devices used for this purpose if rockets and stratospheric spheres are used. We can also create the ideal climate and atmosphere for our lives. Additionally, if hydrocarbons are discovered at a certain depth, we can produce a natural feeding environment for living things. Water under extreme pressure and temperature serves as a source of energy for terrestrial life, a source of oxygen in the atmosphere for settlers' colonies, a fuel for movement, and power for autonomous generators.

According to previous research, in Assessing the potential of Mars to host life and provide valuable resources for future Human exploration, understanding the state of water on Earth is of utmost importance. Therefore, studies have been conducted to determine evidence of the Existence of past or present water on Mars. Although it is widely accepted that abundant water existed very early in the history of Mars, in modern form, only part of this water can be found as ice or trapped inside a structure. Of the abundant water-rich material of Mars. Water on Earth is valued based on various evidence of Rocks and minerals, achondrites from Mars, small temporary salt spills (dune, etc.) Rivers, reactivated gutters, embankment strips, etc., daily flat soil moisture (e.g.) Curiosity and Phoenix Lander), Topographical representation (probably lakes and river valleys), Groundwater, and other evidence collected from the discovery of the spacecraft and rover.

One of the most critical pieces of evidence is related to the ancient riverbed of the Gale Crater, suggesting ancient times. The amount of "strong" water on Mars. Long ago, there was a condition of hospitality for the life of microorganisms. This is because the surface of Mars is likely to be wet regularly. However, the current dry surface makes it stand out. It is almost impossible as a suitable environment for living things. Therefore, scientists Recognized the Earth's underground environment as the best potential place for life research on Mars.

As a result, modern research aimed at discovering central Groundwater. Until NASA discovered a large amount of underground ice and subglacial lakes in 2016 by Italian scientists. However, the Existence of life in the history of Mars is an unsolved problem. In this unified context, the current overview summarizes the results from numerous studies. All relevant discussions on Martian water history explanations and possibilities of the Existence of living things on the Earth.

Moreover, after studying the surface from samples from the rocks and minerals of mars, Most of the water on Mars today is ice, but a small amount of water is ice and Exists as vapor in the atmosphere or as a small amount of liquid brine found in shallow waters floor space. Bright materials that appear to be ice can also be visually seen in the image in a new impact. A high-altitude crater was imaged by HiRISE (High-Resolution Image Science Experiment). On the surface of Mars, water is only visible in the Arctic crown. Elsewhere on Mars. In Antarctica, where there is permanent carbon dioxide, there is a significant amount of water on Flat ground where ice caps and milder conditions predominate. The presence of more than 21 million km3 of ice on or near the surface shows what happens to Mars [10] .

Moreover, for the evidence of mars' surface, As is generally accepted, none of these vast areas of liquid water remain, even though the water was abundant very early in Mars' history. Modern Mars has some of its water. It is clogged with either ice or a rich water-rich substance consisting of sulfates and clay minerals (phyllosilicates) in the structure.

Hydrocarbons and gases have very high resistivity and thin thickness. This means a high degree of resistivity anisotropy. The red polarizability on the left side of the image means low polarizability. The anisotropy blue color on the right side of the photo means a low anisotropy. Also, both blues are expected to mean the minimum cross-sectional area of gas and hydrocarbons, but not water. Red means parts that are likely to be hydrocarbons, and dark blue means parts that are likely to be water. And you can see the big blue one on the left side of all 6 cross sections. It is a geological fault zone. And this zone is changing from upper cross sections to bottom cross sections [11]. And because of the importance of hydrogen in water. So, oxford works on the study of the hydrogen of this water, The majority of the water on Mars is currently found in various reservoirs both below the surface (such as ice polar deposits) and above the surface. The Martian atmosphere also contains a small amount of water, and variations in it are strongly influenced by the seasons [12].

Even though it wouldn't be in equilibrium with the environment, theoretical arguments have been advanced that suggest liquid water could form in transient events.

The current surface pressure on Mars, in particular, has been hypothesized by Kahn [1985] as the result of irreversible carbonate formation in sporadic pockets of liquid water that have occurred periodically throughout Martian history.

A Carbonate formation doesn't stop and the surface pressure doesn't stabilize until the CO2 overburden pressure reaches a limiting value, p *, below which liquid water no longer forms (even in disequilibrium with the environment). According to Kahn, p* is located between water's triple point pressure (6.11 Mbar) and 30 Mbar. So, from above we conclude that the water on mars is available in many ways like vapors or in the liquid state but the major percentage is in the solid state as ice. And there are many studies on the surface of Mars and its mineral. And some of the water samples are taken from the surface to study its component like hydrogen and oxygen which lead us to identify the requirements of life on mars. Phyllosilicates are derived from alteration products of igneous minerals (found in magma) due to long-term contact with water. One example of a phyllosilicate is clay. Phyllosilicates have been detected by OMEGA mainly in the Arabia Terra, Terra Meridiani, Syrtis Major, Nili Fossae, and Mawrth Vallis regions, in the form of dark deposits or eroded outcrops. Hydrated sulfates are formed through interaction with acidic water. OMEGA has detected these in layered deposits in Valles Marineris, in extended surface exposures in Terra Meridiani, and within dark dunes in the northern polar cap. The discoveries have important ramifications for understanding the planet's climate history and if it was once livable. They specifically mention two significant climatic episodes: phyllosilicates first evolved in an early, moist environment, and then sulphates later on in a more acidic environment. Basalt, a fine-grained magmatic rock with a predominance of pyroxene, plagioclase feldspar, and the mafic silicate mineral olivine, makes up the majority of the surface of Mars. These minerals undergo chemical weathering when exposed to water and air gases, which transforms them into secondary minerals. Some of these minerals may mix water with their crystalline structures as hydroxyl throughout the process (OH). Gypsum, kieserite, phyllosilicates (such as kaolinite and montmorillonite), opaline silica, and iron hydroxide goethite are a few examples of hydrated or hydroxylated minerals that have been discovered on Mars. Water and other reactive chemical species are directly impacted by chemical weathering because it consumes, appropriately separates them from the hydrosphere or the atmosphere, and eventually embeds them in minerals and rocks. Although the precise volume of water absorbed by hydrated minerals in the Martian crust is unknown, it is assumed to be rather substantial. For instance, the mineralogical models of the rock outcroppings that were assessed by equipment on the Opportunity rover suggest that deposited sulphate in Meridiani Planum might contain up to 22% water by weight. Every chemical weathering reaction that takes place on Earth involves water to some extent. Although water is necessary for secondary minerals to originate, water is frequently absent from them. Anhydrous secondary minerals include various carbonates, metallic oxides, and certain sulphates like anhydrite. Anhydrous secondary minerals include various carbonates, metallic oxides, and certain sulphates like anhydrite. There may not be a requirement for water, or if there is, it may only be present in very minute amounts, as ice or in very thin molecular-scale coatings, to generate a small number of these weathering products on Mars. It is currently unknown how well such peculiar weathering mechanisms function on Mars. The sort of environment in which the minerals were generated can instead be determined by aqueous minerals, which are minerals that include water or form in the presence of water. Temperature, pressure, concentrations of gaseous and soluble species, and other factors all affect how quickly aqueous reactions happen.

Evidence:

The revolution in ideas about water on Mars was triggered by the Mariner 9 spacecraft in 1971. We have found huge valleys in many areas (Figure 1).Break through the dam and erode the ditch Rock invasion and deep valley carvings are some examples of the physical changes caused by the water attack. Shown in his photo. Considering the area of ​​the branch stream seen in the south.

It increased over time. A map of the 40,000 river valleys on Mars June 2010. This is about four times the number of previously identified river valleys. There are two main classes of floating features on Mars: Seen from a distance Ground and branched Noachian networks. Very long one, A large, insulated, single-threaded drainage channel from the Hesperia era. Some young ones these days Small canals can be seen at mid-latitudes, these are Hesperia Amazon era. These channels can be due to local ice deposits that sometimes melt.

Applying a gamma-ray spectrometer to record the spectra of gamma rays emitted from the Martian surface as the spacecraft passes over different regions of the planet and the Mars Odyssey neutron spectrometer, surface hydrogen has been detected globally in considerable amounts. The molecular structure of ice seems to contain this hydrogen; and based on stoichiometric calculations, in the upper meter of the surface of Mars, some concentrations of ice have been reached by the conversion of the detected fluxes. As a result, ice cream is not only plentiful today, it is also ubiquitous. surface of the planet. It is concentrated in various regions below 60 degrees latitude, including Terra Sabaea, around Elysium Volcanoes, and northwestern Terra Sirenum. It is also present underground, with concentrations containing 18% ice. There is more ice above the 60th parallel, so that its concentration exceeds 25% at the 70th degree line and almost everywhere on earth Paul achieves it 100%. Most of it is presumed because the ice present on the surface of Mars is unstable. It is covered with layers of dust and rock. Observations by the Mars Odyssey Neutron Spectrometer show that the global water equivalent is A layer (WEG) of at least 14 cm (5.5 in) cannot be achieved if the topmost meter of ice on the Martian surface is not evenly distributed. So, on average, the surface of Mars is about 14% water. Also, the poles of Mars correspond to a WEG of 30 m (98 ft).About the geology of the planet Historically, geomorphological evidence shows large amounts of surface water with WEG up to a depth of 500 m. (1600 feet). Even if the detailed mass balance of the controlled process is still unknown Some of this past water may have been lost deep underground or into outer space. Gradually ice Moved from one part of the Earth's surface to another. This happened seasonally and over time It was a time scale and was supported by a modern atmospheric reservoir. But in less ways Volume is not important if it is larger than 10 micrometers (0.00039 inches) [150]. NASA's Mars Exploration Program has prepared Groundwater Ice Mapping (SWIM). (Fig. 9) For North Hampshire on Mars. In Figure 9 the blue shaded area indicates where the data is located consistent with the presence of subsurface ice while red indicates discrepancies between them Dates and the presence of underground ice. Each ice-exposed impact site corresponds to a region It is estimated by the SWIM equation to be consistent with the presence of groundwater ice. The seasonal behavior of Martian water is an important indicator of many aspects of the Martian surface and atmosphere composition. Perhaps the most interesting area is the current Martian climate and how it has changed over geological time. By studying the variability of atmospheric water, the behavior of polar ice caps and their seasonal cycles, the exchange of volatiles between the atmosphere and subsurface regolith, and the possible consequent chemical processes, dynamics, and atmosphere Mechanisms that may result in the overall circulation of and the generation of net sources or sinks of volatiles throughout the year. By understanding the physical processes that occur over a year, we can begin to understand the climate history of Mars by studying its response to the long-term effects of changes in the orbits of the Sun and Mars. There is ample evidence of dramatic climate change or variability on Mars. Perhaps most interesting is the existence of channels carved by liquid water at a time when surface water was more stable than it is today. The stratified topography of the polar region also suggests dramatic changes, apparently mixing layers of water, ice and dust in varying amounts, suggesting the variability and temporal nature of transport of both species to the poles. Is shown. Nitrogen isotope measurements show that the atmospheric abundance of nitrogen has changed significantly, whereas oxygen isotope measurements require a significant long-term exchange of atmospheric oxygen with non-atmospheric sources (McElroy et al., 1976; Nier et al., 1976; Fanale and Cannon, 1978). Variations in the orbital inclination of Mars occur periodically (e.g., Ward, 1979), and there is a correlation between climate change caused by changes in inclination and the formation of polar landforms, and possibly by watercourses, and the evolution of Mars' composition. Suggesting causality. It creates an atmosphere. This paper reviews observations of atmospheric water vapor and its variability over seasonal cycles and discusses physical processes that may contribute to the variability. Results are integrated with other data to provide an internally consistent picture of current climate overviews and seasonal water cycles. That is, the column frequency of water vapor in the atmosphere changes Near zero to about 100 precipitable microns (100 pr txm, equal to 10-2 g H2O cm-2), depending on location and time of year. Globally integrated atmospheric abundances vary between 1 and 2 x 1015 g (Jakosky and Farmer, 1982). Many processes contribute to this variability. Movement of the atmosphere Redistribute water seasonally and replace it with steamless reservoirs. Contributes to both local and global fluctuations. Possible seasonal reservoir contained in water Ground in a non-polar location at some time of the year or day. Subsurface ice in equilibrium with atmospheric vapor or in diffusive contact with the atmosphere On a timescale much less than a Mars year; adsorbed water molecules bodily connected to the regolith profits and in diffusive touch with the surroundings on a timescale shorter than a Mars year; subsurface liquid water, both fashioned domestically as a temporary state among melting of ice and evaporation or in equilibrium with the atmospheric vapor because of the melancholy of the freezing factor and vapor stress because of the presence of dissolved salts; or chemically-sure water that exists as water of hydration of floor minerals. Each of those reservoirs for water may be thermally activated, so the seasonal versions of floor and subsurface temperatures can pressure water vapor into or out of the surroundings. Additionally, water may be gifted withinside the surroundings in non-vapor states, withinside the shape of cloud debris or adsorbed onto airborne dirt grains; shipping in those states can be vital withinside the worldwide redistribution of water. Most of the alternate approaches cited are, in all likelihood, vital at a few locations and a few seasons. The predominant thrust of this evaluation might be to look at the evidence referring to every manner and to decide if their relative significance may be delineated. One of the main goals of this body of research is to understand transport. How much water occurs in a Martian year and relate this to different processes, Occurrence and effectiveness of interaction with each reservoir. The process can be Extrapolated to other times to understand the modern history of Water exchange between reservoirs. This result is important for testing the formation and development of watercourses or polar stratified landforms, For example, or the development of the climate as a whole. The next section discusses the physics of equilibrium between water vapors. Moreover, the condensed phase including modification by water adsorption and the Presence of dissolved salts. This understanding will be used in subsequent discussions. Exchange of water between atmospheric reservoirs and surface or underground reservoirs. Section 3 analyzes seasonal water cycle observations. Related Observations include the frequency of water vapor plumes as a location function. Season, vertical distribution and saturation of water vapor in the atmosphere, the formation and existence of clouds. The physics of polar caps will be discussed in Section 4. caps are important For two reasons:

(1) the behavior of the remaining polar caps ultimately controls the whole. Circulation of water vapor due to its ability to act as a source or sink of water vapor.

(2) Since water ice is included in the seasonal ceiling together with CO 2 frost, Caps represent potentially important seasonal reservoirs.

ii. The atmosphere

By electrolyzing simulated Martian regolith brine (SMRB), one of the demonstrated methods to help the possibility of life on other planets was directed toward Mars. The research suggested conducting numerous tests to identify the best materials to use in order to produce an acceptable outcome. Despite being effective, the technique had a weakness given the volume of the material used. There are many tools available to guide the search for water. The NASA Phoenix lander used these instruments to discover evidence of a functioning water cycle, a significant amount of subsurface ice, and the presence of soluble perchlorates on the Martian surface. Large amounts of water ice are present in the northern polar region of Mars, according to additional spectral data from the Mars Odyssey Gamma Ray Spectrometer, and the Mars Reconnaissance Orbiter has also discovered evidence of recent local flows of liquid regolith brines forming Martian geography [9]. All of these resources were essential for taking the various types of research for this purpose further. The electrolysis of SMBR was the method's intended target. To find the materials that would produce the highest efficiency, numerous tests were conducted. High-performance alkaline water electroliers using Pb2Ru2O7- as oxygen evolution reaction (OER) electrocatalysts were one method suggested based on the concept of OER. It is well known that such electrocatalysts are active in both oxygen reduction and evolution reactions. The electrolier’s anode was made of Pb2Ru2O7-. Initial experiments in an O2-saturated SMRB showed negligible faradaic contributions from the base Glassy Carbon (GC) electrode and a slight increase in current vs. Ag wire at potentials over 1.3 V. In contrast to RuO2, which showed OER activity at potentials over 0.9 V vs. Ag wire, Pb2Ru2O7 was OER-active even at 0.1 V vs. Ag wire. The results showed that Pb2Ru2O7 had significantly more simple OER kinetics in SMRB when compared to RuO2 and GC. The literature has reported that Pb2Ru2O7 has an OER pathway that is involving a higher oxidation state of Ru is recognized as the potential determining step, and it is mediated by a Ru-active site. With a higher Ru(V):Ru(IV) ratio than RuO2, Pb2Ru2O7 displayed greater OER activity because the higher oxidation state of the Ru intermediate was stabilized, lowering the OER activation barrier. To mimic the Martian environment, several tests were conducted at various temperatures. The plot of linear sweep voltammograms (LSVs) of Pb2Ru2O7-pyrochlore, RuO2, and GC in 2.8 M magnesium perchlorate at (A) 21 °C and (B) 36 °C is shown in the following figure. (C) OER current density for Pb2Ru2O7-pyrochlore, RuO2, and GC under an O2-purged environment over a temperature range (21 °C [leftmost point] to 36 °C [rightmost point]) vs. OER onset potential. At pH = 0, 25 °C, and atmospheric pressure, E0 (Ag wire) = 0.44 V vs. SHE. [16] In terms of energy efficiency, varying the voltage applied to the electrolier at a temperature of 50 °C produced a number of satisfactory results. It was discovered that the electrolier had a faradaic efficiency of about 70 \%, a voltage efficiency of 68–100 \%, and energy efficiency of 36–60%. Because O2 can become the dominant gas in the atmosphere A lifeless planet, by itself, is not robust biometric signature. Our results do not necessarily preclude its usefulness for now. But they show that the situation is Much more complex than previously thought May create an abiotic O2-rich atmosphere Complex features of planetary accretion histories, internal Chemistry, atmospheric dynamics, orbital conditions. detection Range of possible terrestrial planets with variables N2 and Ar reservoirs should be a rich area for future theoretical investigations Research that helps us better understand climate evolutionary mechanism. Nevertheless, for some exoplanets, Detailed modeling may not lead to definitive conclusions Given the uncertainty inherent in the process such as volatility Delivery at the time of incorporation. Observing, there may still be a way to tell the difference Scenarios are described here, but only if a reliable method is developed Get the O2 and N2 or Ar ratio of an exoplanet atmosphere. In principle, this can be done by analyzing Spectrally resolved planetary phase curves , During transport by measuring the spectral Rayleigh scattering Suspended in a cloud-free (i.e., aerosol-free) atmosphere or possibly by spectroscopy Characterization of oxygen dimers. I will have more work necessary to determine the feasibility of these technologies O2/N2 mixing ratios in realistic planetary atmospheres. How would things change on a more complex planet Were other atmospheric components present? First, if the atmosphere contains N2 or Ar, the required amount of O2 Blocking H2O outlets is greatly reduced and increased Horizontal heat transport reduces the likelihood of atmospheric bistability due to O2 condensation in the region of the planet Low surface setting. reducing gases such as methane, Can be outgassed from planetary interiors by abiotic processes, might have The variation in O3 and NOx concentrations is A function of UV level and atmospheric composition change this. In addition, sulfur species emitted from volcanoes and heterogeneous chemistry also affect this Atmospheric redox balance. In future investigations using photochemical models, These effects as a function of water loss rate. Surface/interior redox exchange is another source of complexity for Earth-like low-N2 planets. when planets form The hydrogen shell is lost to space early, and its crust and oceans may be the first to deplete and react with the oxidation products of HO photolysis. Quick surface first. But as long as this is happening, the upper atmosphere will remain H2O rich and fast Photodegradation may continue. Over time, the surface of the planet and The inside is oxidized, and the ability to act decreases. as an oxygen sink. Given her XUV flux on Earth today, is the lower bound of her H2 escape from the hydrogen rich homopause. 4 × 1010 molecules cm−2 s−1. given that Soil low in nitrogen can lose 2.1 x 1022 mol of H2O 4 Gy or 28% of current ocean volume 7. 66.2 bar Atmosphere O2-Sufficient amount to cause it Significant reversible oxidation of solid planets, and therefore Significant reduction in reducing power on the surface. There XUV flux is significantly enhanced around young dwarfs In general, total water loss can be many times this value. Case. After all, planets like Earth can emit CO2. Potential of plate tectonics and carbonate silicates Weathered feedback. The CO2 cycle is spinning The first anoxic planets that do not contain N2 or Ar are complex. CO2 condenses at relatively high temperatures, but solids have lower compressive strength. No ocean / internal heat transport The process can accumulate outgas CO2 in low plants. The region of the planet until the backflow is from the CO2 glacier Enough to return to high deployment areas. Apart from the complexity of the overall climate problem However, future investigations can show the possibility of this. The earth has a very high CO2 level It has a global average surface temperature similar to that of the earth. [12] The $O_2$ buildup mechanism can easily be understood intuitively by a thought experiment involving a hypothetical planet with a pure $H_2O$ composition. Lacking atmospheric N2, Ar, and CO2, such a planet will ini￾tially have a pure H2O atmosphere, with the surface pressure determined by the Clausius–Clayperon relation (Andrews 2010; Pierrehumbert 2011). If the planet has the same orbit and inci￾dent stellar flux as present-day Earth, it will most likely be in a snowball state (Budyko 1969). However, because H2O cannot be cold-trapped when it is the only gas in the atmosphere, it will be photolyzed by XUV and UV radiation from the host star (primarily via H2O+hν → OH∗ + H∗). The resultant atomic hydrogen will escape to space at a rate dependent on factors such as the XUV energy input and the temperature of the ther￾mosphere, and hence the atmosphere will oxidize.2 In the one-dimensional limit with no surface mass fluxes, atmospheric O2 will build up on such a planet until pn is high enough to cold-trap H2O and reduce loss rates to negligible values. In three dimensions, the initial atmospheric evolution may depend on the planet`s orbit and sub-surface heat flux/ transport rate, because on a tidally locked, ice-covered planet with pure H2O atmosphere, conditions on the dark side could be so cold that even O2 would condense. But on one planet Earth-like rotation and tilt, all regions of the planet You can see the starlight once a year, so once on the surface of the earth O2 The inventory has exceeded a certain threshold and an O2 atmosphere is being built Probably inevitable. Also, temporarily for each planet Warming events such as meteorite impacts can be forced When the atmospheric pressure is high, it transitions to the steady state. 3 What about more general scenarios? First, you can relax it Infer assuming zero downflow on the surface A case where the generated O2 can be used for internal oxidation. Next, the redox balance determines the oxygen content in this atmosphere. Must accumulate until the loss of hydrogen to space is compensated Surface removal rate of oxidizing substances. for example, $O_2$ removal rate of $5 \times 10^9$ molecules $cm^{-2}s^{−1}$ on the surface 4 Diffusion limited in case of leakage he is balanced by $H_2O$ loss.

iii. Another Proxy

The presence of Oxygen on a surface planet has been considered as a signature of existence of living organisms due to the fact that Oxygen is produced as a result of some form of plants which are producing it through photosynthesis. This was the case until a research team led by Dr. Narita has shown that abiotic oxygen produced by the photocatalytic reaction of titanium oxide, which is known to be abundant on the surfaces of terrestrial planets, meteorites, and the moon in the solar system, cannot be discounted. According to NASA, stating in an article on September 9, 2015, the article proposed the following: [13]

“For a planet with an environment similar to the sun-Earth system, continuous photocatalytic reaction of titanium oxide on about 0.05 \% of the planetary surface could produce the amount of oxygen found in the current Earth's atmosphere. In addition, the team estimated the amount of possible oxygen production for habitable planets around other types of host stars with various masses and temperatures.” In that case, the presence of oxygen on an exoplanet is not the only factor that indicates life signatures, but rather it can be considered as a proxy to indicate other additional biosignatures. This studying of Dr. Narita – and any other researches done that took the same approach – has led to applying the concept of ‘False-positive case scenarios’ to the cases where oxygen is present on another planet. Because of that, another method of detection has been proposed in this research, in addition to another proxy that can be more depended-on than oxygen to ensure life. [14]

iv. Chemical Disequilibrium

An unusual method that has been used to detect biosignatures on exoplanets was by detecting the chemical disequilibrium of these planets. Although this method was exposed to several arguments, it provided a merely trusted probability of existence of life on a planet when compared to Earth’s disequilibrium. [15] Because of the agreement to the fact that biosignatures of abiotic oxygen on exoplanets has been leading to “false-positive scenarios” in most cases, the scientific community started to construct researches to detect other possible, more-believed biosignatures to indicate life on exoplanets. A study has chosen the proxy to be related with a kind of environmental disequilibrium resulted from chemical conditions on a planet. A group of scientists tried to prove how helpful might that help by comparing the chemical “especially thermal” disequilibrium of some eras of Earth to other planets’ ones. [15] The model constructed depended on calculating the available Gibbs free energy from a selected era until the modern time. The general formula proposed were:

$$\Phi \equiv G_{(T,P)}(n_{initial}) - G_{(T,P)}(n_{final})$$

Figure 12. Available Gibbs free energy from an era to the modern time

The only flaw that comes with this method of detection is stated as follows: “the magnitude of atmospheric disequilibrium does not—on its own—indicate the presence of life. Further interpretation is necessary.” Meaning that additional factors are still required to ensure not falling under false-positive scenarios. [15]

v. MOXIE

FM OC9 to OC13 runs were completed during a semiannual high-density season (northern hemisphere spring). FM OC14 occurred as air density was dropping, and FM OC15 occurred near the yearly minimum density. The generic runs OC10, OC11, OC14, and OC15 were largely identical except for the time of sol and the seasonal atmospheric density. Generic MOXIE runs begin with a reference segment described by McClean in which MOXIE's compressor is commanded to a revolution per minute calculated to input 55 g/hour of martian atmosphere into the system (55 g/hour is the maximum reliable intake during annual atmospheric density minimum with MOXIE's compressor at its maximum of 3500 rpm). The MOXIE system supports the option of choosing between specifying the desired voltage across the top and bottom portions of the stack or specifying the desired current through the stack (current-control mode, which effectively controls the oxygen generation rate). In order to get the necessary current, a feedback loop changes the stack voltage during each run detailed in this work. [16]

III. Conclusion

To begin, life is complex, and humans cannot meet all of its demands. So, in order to exist on another planet, you must first have a site like Earth that is in the habitable zone (HZ). As a result, a hapitable zone is described as a region that includes stars, atmosphere, and water. Thus, liquid water is the desired outcome. And after much research, we discovered that Mars possesses water in every state (like earth). However, the biggest disadvantage in the solid state as ice. That was due to the low pressure. [9] However, according to "NASA," there is just a small amount of the liquid. And the atmosphere on Earth is made up of a variety of chemical substances in the form of gases. In addition, to replicate this climate and atmosphere. "NASA" is developing a model to serve as an atmosphere. Additionally, ocean circulation controls the distribution and activity of life on Earth as well as the interaction between life in the water and life in the environment. Ocean circulation plays a crucial role in both the oxygenation of our planet and the search for extraterrestrial life because it eventually restricts the development of biological products in planetary atmospheres. We used a global ocean atmosphere GCM to investigate ocean dynamics and the consequent ocean ecosystems on worlds other than Earth.

I. Introduction

Coronavirus was first identified in 1968 by a group of scientists who conveyed their findings to the journal Nature. Since the virus was pointed out under an electron microscope, it looked as a circular disk. Because of its shape, they have chosen the name Corona, which meant the sun. After various intensive research, it was discovered that Coronaviruses are single-stranded RNA viruses. Furthermore, they have a great susceptibility to mutate and recombine leading to highly diverse copies. These diversities allowed them to have 40 distinct varieties which permitted infecting bats and wild animals. Therefore, Chinese populations who feed on wild animals’ meat, have been contaminated, which resulted in the whole world infection. In consequence, the number of death people reached about 433,107, that stimulated researchers to find anyway to diminish the virus. For example, transmitting plasma –occupies about 55% of blood components- from cured patients. That is due to the presence of emerged immunoglobulins (antibodies) from the immune system that defeated the virus. Unfortunately, this is a high-cost, inefficient and time-wasting process, as it is not applicable to all the recovered patients and requires exorbitant medical appliances. Regarding these disadvantages, this research concerns using a new divergent method to deal with Coronavirus using gene expression regulators called non-coding RNA (ncRNA), that is represented in microRNA (miRNA), piwi-interacting RNA (piRNA) and short interfering RNA (siRNA). Since this RNA is non-coding, so it could be synthesized and injected to infected humans without significant interaction or interfering with the body’s internal regulation.

II. Viruses

From scratch, it is useful to have a glance about viruses, their discovery, composition, classification, and replication cycles. Actually, scientists could detect viruses before they could see them. It goes back to the Tobacco mosaic disease that impedes the growth of the plant and gives its shape a mosaic coloration. Scientists noticed that it could be transmitted simply by rubbing sap extracted from diseased leaves onto healthy ones. It was hypothesized that there was an infectious invisible bacterium that was the intrinsic reason. In further experiments, it was found that the pathogen could replicate only within the host. Moreover, it couldn’t cultivate on nutrient media in petri dishes or test tubes. Martinus Beijerinck, the Dutch botanist who was working on a series of experiments, was the first scientist to announce the concept of a virus. In 1935, it was confirmed by the help of electron microscope that the infectious particle was a virus and now called tobacco mosaic virus (TMV).

III. Results

After the basic information about viruses in the prior paragraphs, it’s the time to talk about COVID-19 in a specific term and to compare its composition and genome with the other members in the Corona family of viruses. Coronaviruses are members of the subfamily Coronavirinae which in the family Coronaviridae and the order Nidovirales. the recent outbreak of unusual coronavirus pneumonia in China and then worldwide pandemic is 2019-nCoV or COVID-19. According to its genomic structures and phylogenetic relationships, COVID-19 belongs to the Beta-coronavirus genera that has a very close similarity of the sequences of COVID-19 and that of severe acute respiratory syndrome-related coronaviruses (SARS-CoV) mentioned in (Table 1). As a result of this closeness of the SARS-CoV-2 and that caused the SARS outbreak, the Coronavirus Study Group of the International Committee on Taxonomy of Viruses called the virus as SARS-CoV-2.

A Usual Coronavirus contains six ORFs in its genome at least. Gammacoronavirus is an exception as it lacks nsp1, the first ORFs (ORF1a/b), that is two-thirds of the entire genome length, encodes 16 nsps (nsp1-16). Both of the ORF1a and ORF1b contain a frameshift which produces two polypeptides: pp1a and pp1ab. These two polypeptides are developed by an encoded virally chymotrypsin-like protease (3CLpro) or a main protease (More), in addition to one or two papain-like proteases into 16 nsps. In which all the accessory and structural proteins are translated from the sgRNAs of Coronaviruses. The four main structural proteins contain membrane (M), spike (S), nucleocapsid (N) and envelope (E) are encoded by ORFs 10, 11 on the one-third of the genome itself, near the 3-terminus. By the way, different coronaviruses codes special accessory and structural proteins. For example, 3a/b protein, HE protein, 4a/b protein as shown in (Fig. 5B). they are responsible for various essential functions in viral replication and genome maintenance. The most plentiful structural protein is M glycoprotein which spreads over the membrane’s bilayer for three times, leaving NH2-short terminal domain outside the virus and a COOH long terminus that is a cytoplasmic domain inside the virion. Also, it plays sub-dominant role in the in the intracellular origination of viral particles without requiring S protein. Additionally, the S protein as a kind of 1 membrane glycoproteins that compromises the peplomers. Actually, it is the main inducer of neutralizing antibodies, between the E proteins, which subsist a molecular interaction that determines composition and formation of the coronavirus membrane. Moreover, in the existence of tunicamycin -a mixture of antibodies that inhibit UDP-HexNAc, the polyprenol-P HexNAc-1-P family of enzymes- CoVs grows with a lack of S protein that is noninfectious virions with only M protein. During analyzing genomic structures of viruses, it was found that the 5-end UTR and 3-end UTR are involved in intra and intermolecular interactions. As well, are crucial for RNA-RNA interactions also for binding viral and cellular proteins. As shown in (Fig. 6), from the beginning of the 5-end side, Pb1ab is the first ORF of the entire genomic length, encodes non-structural proteins of 29751bp (7073aa), 29844bp (7096aa) and 30119bp (7078) in SARS-CoV, COVID-19 and MERS-CoV, respectively. On the other hand, the S protein at 3-end in comparison between the three viruses, are 21493aa, 1273aa and 1270aa for SARS-CoV, COVID- 19 and MERS-CoV, respectively. Finally, it is concluded that COVID-19 is less similar to MERS-CoV and SARS-CoV with a percentage of 50% and 79% respectively, as the arrangement of the dominant structural proteins N, S, E, and M are different as demonstrated in (Fig. 6).

IV. Non-Coding RNA (Micro-RNA)

Various intensive studies on the human genome, especially on the transcriptional landscape, have shown that only 2% of human genome is for protein translation. What about the rest of the genome that reaches about 98%, what they are synthesized for? Actually, this is one of the fundamental purposes of the paper. It was found that the rest of the genome was entirely non-coding sequences. Considering the fact that every particle in the human body was created for a particular beneficial function, these remaining sequences should have essential functions. They are called non-coding RNA (ncRNA), as a result that they don’t take part anyway in protein translation. They bind to distinct types of DNA and RNA in different specific ways, leading to alternations in their transcription, their degradation, editing, processing and aids in translation. In fact, they form complex regulatory networks due to their competing for binding to mRNAs, other may call them competing endogenous RNA (ceRNA). They are classified into two major classes based on transcription length. Where small ncRNAs (≤200 nucleotides) include small interfering RNA (siRNA), microRNA (miRNA) and PIWI-interacting RNA. While those which are >200 nucleotides, are called long ncRNA (LncRNA). Comparing mRNA, LncRNA lack ORFs and are mostly specific to deal with tissues. Taking that in consideration, means that we will talk about miRNA as it deals in the cell level. The discovery of miRNA dates back to 1993 when a group of researchers called Ambros and Ruvkun groups. They were working lin-4 that is a one of the genes that regulate temporal development of C. elegans larvae as well as the other gene lin-14. they found that lin-4 wasn’t a protein coding RNA but a small noncoding RNA. Since that fact lin-14 was post- transcriptionally down-degraded by its 3-end untranslated region (UTR) and lin-4 had a complement sequence to that of the 3-end UTR. by the way, it is proposed that lin-4 regulates lin-14 in the post-transcriptional level. miRNA averages 22 nucleotides in length. Where in most cases, they interact with the 3-end UTR of target miRNAs to suppress and degrade their expression. However, its interaction with other regions like 5-end RNA is also required. Furthermore, it was recently reported that they control the rate of translation and transcription as they are shuttled among subcellular compartments. The biogenesis process of miRNA is more distinct and convoluted from that of the other ordinary types of RNA. This process is classified as canonical and non-canonical pathways. For the canonical biogenesis: it is the dominant pathway in which miRNA is synthesized. pri-miRNA begins by transcription from their initiate genes, followed by processing into pre-miRNA by the micro-repressor complex that is composed of an RNA binding protein DiGeorge Syndrome Critical Region 8 (DGCR8) which recognizes an N6-methyladenylated GGAC and other motifs in primiRNA and Drosha, a ribonuclease III enzyme that cleaves the primiRNA at the base of the hairpin structure. This leads to the formation of a 2 nt 3-end extended over pre-miRNA. After premiRNA is resulted, it is exported to the cytoplasm by a complex called exportin 5 (XPO5)/RanGTP. Followed by processing by the RNase III endonuclease Dicer. These series of steps involve the terminal loop removal, leading to a miRNA duplex as shown in (Fig. 8). The 5p strand originates from the 5-end of the resulted pre-miRNA hairpin while that of the 3p arises from the 3-end. After that, both of the strands are loaded into the Argonaute protein (AGO) in an ATP-dependent manner. Moreover, the proportions of AGO loaded 3p or 5p strand given for any miRNA varies depending on the cell type and environment. Where the selection of the 3p or 5p strand is partially based on the thermodynamic stability of the 5-ends of miRNA duplex or 5-end uracil (U) -a special nucleotide that is present only in RNA instead of adenineat nucleotide position 1. Accompanied with more illustration, the strand 5-end stability or 5-end U is loaded into AGO and is considered the guide strand. So, the unloaded strand is called the passenger one, that is unwounded from the guide strand through various mechanisms according to the degree of complementarity. The passenger strand that contain mismatches, are split by AGO and degraded my internal machinery to produce a strong bias, but if they are non-AGO2 loaded or have central mismatches, they are unwound and degraded. The silencing miRISC complex’ formation begins with recruiting a family of proteins by miRISC called GW182. This family provides the needed scaffolding to recruit further effector proteins. For example, CCR4-NOT and PAN2-PAN3 poly A deadenylase complexes. This follows miRNA and targeted mRNA interaction. With taking in consideration, target mRNA poly A deadenylation is instituted by PAN2/PAN3 and is completed by CCR4-NOT complex. Where the interaction between the tryptophan W -repeats of GW182 and poly A-binding protein C (PABPC) stimulates for efficient deadenylation. Moreover, decapping happens simply by decapping protein 2 (DCP2) and the other associated proteins. Followed by the irrevocable step of 5- end to 3-end degradation by the action of exoribonuclease 1 (XRN1) as illustrated in the final step of (Fig. 8).

V. Technologies and Techniques to be applied

VI. Conclusion

In conclusion, treating COVID-19 using miRNA is a new distinctive technological method affords to attack the virus once injected in infected human bodies, thanks to the guiding proteins for miRNA strands. There is no doubt that this field needs more intensive research to be clearer, but applying this method, may be used to encounter COVID-19 as we as other RNA-Viruses. Furthermore, focusing on non-coding RNA, may allow the humanity to have a secondary immunity against incoming viruses.

I. Introduction

By default, human beings are social creatures who need to belong to a community and a society. Right at the center of Maslow’s Hierarchy of Needs, love and belonging stare at lone wolves mocking the “I can do it alone” myth they are convinced of. For the sake of belonging to a group, people endure various sacrifices from eating certain foods to acquiring different attitudes. The need for acceptance and attachment intensifies during adolescence and may increase the amount of control exerted by a peer group on an individual. Being under the control of a peer group causes peer pressure, which can be defined as group insistence and encouragement for an individual to be involved in a group activity in a particular way (Anon, 2012). Peer pressure occurs when an individual experiences persuasion to adopt values, beliefs, and goals of others (Feldman, 2011; Wade, 2005). It is likely that peers influence school engagement; for example, peers may positively reinforce conventional behaviors, exert pressure toward school involvement, or model positive affect and commitment to academic endeavors (Fredricks, 2005). The importance of positive peer pressure is clear in different studies showing that students who are persuaded verbally that they can carry out academic tasks show effort and perseverance, at least temporarily, when faced with challenging tasks (Klassen, 2010). Teenagers seek out friends who engage in similar activities (Suls, 2003) and who appear to be their role models (Maibach et. al., 1995). In turn, they are influenced by such activities (Bruess et.al., 1989; Neil et. al., 2007). The main goal of this study is to gain a better understanding of the relation between peer pressure and self-efficacy. The great influence peers have on each other exists due to the importance of peers in one’s life, but it is not the only force on the battlefield. After all, teenagers are not hypnotized into doing only what others around them do. If teenagers are already confident and certain about what is appropriate and correct, then others’ behavior will be largely irrelevant and thus not influential. The concept of self-confidence connects to Bandura’s Theory of Self Efficacy. Self-efficacy refers to belief in one’s capabilities to organize and execute the courses of action required to produce given attainments (Bandura, 1997). That is, self-efficacy belief allow someone to answer the question “Can I do this?” (Hodges, 2008). A glimpse of what self-efficacy represents to humanity is Peterson and Arnn’s argument that self-efficacy is the foundation of human performance. Children who receive negative feedback all the time are more likely to show lower performance because they have been convinced that they are “less capable.” People with high assurance in their capabilities, gained from family and society, approach difficult tasks as challenges to be mastered rather than as threats to be avoided (Bandura, 1994). Bandura focuses on the role of that efficacious outlook to produce personal accomplishments, reduce stress, and lower vulnerability to depression. In his theory, he puts forward four pillars to building self-efficacy, one of which is social persuasion. “People who are persuaded verbally that they possess the capabilities to master given activities are likely to mobilize greater effort and sustain it than if they harbor self-doubts and dwell on personal deficiencies when problems arise”. When surrounded by more advanced peers, the need for social persuasion is intensified. We tend to compete in the highly unhealthy competitive environment to feel safe. Some are ahead of the race and some are not; those who are not challenge themselves with what their peers are capable of and they are not. When the results of these extremely hard challenges are not satisfying, Bandura’s first component of self-efficacy, Mastery Experiences, is jeopardized. In prior research it has been suggested that adolescent self-efficacy is significantly influenced by peers (Schunk & Meece, 2005). Schunk discussed that the perception of peer pressure is inversely proportional to adolescents’ self-efficacy. In their study, students with a score indicating a low level of peer pressure had significantly higher academic self-efficacy scores than did students with scores at moderate and high levels of peer pressure. Adolescents can increase their self-efficacy expectations if school psychologists/counselors work with them to teach them how to resist and properly handle peer pressure (Anon, 2012). However, access to proper psychological care is not available for everyone, especially in developing countries and to people with low socioeconomic status. A factor of great influence here is self-awareness. “That the birds of worry and care fly over your head, this you cannot change, but that they build nests in your hair, this you can prevent.” - Chinese Proverb. It is true we do not control how peers interact with us but being self-aware reduces the intensity and provides justification of why and how adolescents feel the way they do or act the way they act. By definition, to be self-aware is to be able to focus on your actions, feelings, and principles (Alfredsson, 2021). Those vulnerable to peer pressure see themselves through the eyes of others. They do not fully comprehend what they are or how they work. Alfredsson argues that people with a strong sense of how and why they behave have greater self-efficacy and are generally more psychologically healthy. Referring to all the previous arguments, this study was designed to test whether there is a negative correlation between self-efficacy and peer pressure. The relevance of self-awareness to both self-efficacy and peer pressure was tested with the prediction of a positive correlation between awareness and self-efficacy.

II. Method

Participants

Measures

Three main variables were set to test in this study. We measured self-efficacy, peer pressure, and self-awareness by quantitative means via three different questionnaires shared with participants on the internet using Google Forms. The various questions were chosen from the three questionnaires to create the new, unique questionnaire of this study. We added some demographical questions as well asking for age, gender, and educational levels.

Peer Pressure

Peer pressure was measured by the Peer Pressure Questionnaire PPQ (Darcy et. al. 2000). It was measured on a scale of 1-5, one meaning strongly disagree and five meaning strongly agree. Sample questions are “It is usually hard to say no to other people.” and “At times, I’ve broken rules because others have urged me to.” The scale used is an 8-item measure.

Self-efficacy

Self-efficacy was measured by the General Self-Efficacy Scale GSE (Ralf et. al., 1992). It was measured on a scale of 1-4, one meaning strongly disagree and four meaning strongly agree. Sample questions are “It is easy for me to stick to my aims and accomplish my goals.” and “I am confident that I could deal efficiently with unexpected events.” The scale used is a 7-item measure.

Self-awareness

Self-awareness was measured by the Self Reflection and Insight Scale SRIS (Anthony, 2022). It was measured on a scale of 1-6, one meaning strongly disagree and six meaning strongly agree. Sample questions are “I rarely spend time in self-reflection.” and “ I don't really think about why I behave in the way that I do.” The scale used is an 8-item measure.

Procedure

Participants completed a Google Form through the internet, shared via WhatsApp and Telegram by the PI and asked friends to share it around. We integrated the questions from three different questionnaires. No time or place restraints were set. The average completion time of the questionnaire was 3-4 minutes. Participants answered demographic questions about gender, education level, and age. The second section of the questionnaire was an 8-item measure of peer pressure (Darcy et. al., 2000). The third section was a 7-item measure of self-efficacy (Ralf et. al., 1992). The fourth section was an 8-item measure of self-awareness (Anthony, 2022). Participants then submitted the form but received no monetary compensation for their time.

Design

The design of the study aimed at discovering the correlation between peer pressure and self-efficacy, and whether self-awareness could be considered a protective measure against potential negative effects of peer pressure. This study implements a convenient sampling method, where all participants are chosen randomly from a population of Egyptian adolescents. Quantitative data was collected and analyzed using SPSS version 26 program to investigate and visualize the relation between raw scores of different participants.

III. Results

As shown in Table 2, there is a positive correlation between self-efficacy and peer pressure r = .112, p = .153. There also a positive correlation between self-efficacy and self-awareness r = .158, p = .081.

		Age	Mean of Peer Pressure	Mean of Self Efficacy	Self-awareness Mean
Age	Pearson Correlation	1	-.233*	.053	.225*
	Sig. (1-tailed)		.019	.320	.022
	N	80	80	80	80
Mean of Peer Pressure	Pearson Correlation	-.233*	1	.116	-.014
	Sig. (1-tailed)	.019		.153	.451
	N	80	80	80	80
Mean of Self Efficacy	Pearson Correlation	.053	.116	1	.158
	Sig. (1-tailed)	.320	.153		.081
	N	80	80	80	80
Self-awareness Mean	Pearson Correlation	.225*	-.014	.158	1
	Sig. (1-tailed)	.022	.451	.081
	N	80	80	80	80

Table 3 illustrates the positive correlation between peer pressure and self-efficacy, r = .122, p = .299 in males as opposed to a weaker r = .097, p = .232 in females. There is a significant positive relation between self-efficacy and self-awareness r = .613, p = .002 in males as opposed to an r = .005, p = .486 in females. Age appears to be inversely proportional to peer pressure with an r = -.139, p = .274 for males and an r = -.289, p = .013 in females.

Sex			Mean of Peer Pressure	Mean of Self Efficacy	Age	Self-awareness Mean
Male	Mean of Peer Pressure	Pearson Correlation	1	.122	-.139	.271
		Sig. (1-tailed)		.299	.274	.117
		N	21	21	21	21
	Mean of Self Efficacy	Pearson Correlation	.122	1	.194	.613**
		Sig. (1-tailed)	.299		.200	.002
		N	21	21	21	21
	Age	Pearson Correlation	-.139	.194	1	.293
		Sig. (1-tailed)	.274	.200		.099
		N	21	21	21	21
	Self-awareness Mean	Pearson Correlation	.271	.613**	.293	1
		Sig. (1-tailed)	.117	.002	.099
		N	21	21	21	21
Female	Mean of Peer Pressure	Pearson Correlation	1	.097	-.289*	-.128
		Sig. (1-tailed)		.232	.013	.167
		N	59	59	59	59
	Mean of Self Efficacy	Pearson Correlation	.097	1	-.024	.005
		Sig. (1-tailed)	.232		.428	.486
		N	59	59	59	59
	Age	Pearson Correlation	-.289*	-.024	1	.195
		Sig. (1-tailed)	.013	.428		.069
		N	59	59	59	59
	Self-awareness Mean	Pearson Correlation	-.128	.005	.195	1
		Sig. (1-tailed)	.167	.486	.069
		N	59	59	59	59

IV. Discussion

This study was conducted to shed light on the kind of influence peers have on each other’s self-efficacy. In terms of peer pressure, self-efficacy, and self-awareness, we aimed at exploring the problems facing the youth in various educational levels. Results show an overall slight positive correlation between self-efficacy and peer pressure. As per Bandura’s Theory of Self-efficacy, it is illustrated that people seek proficient models who possess the competencies to which they aspire. Through their behavior and expressed ways of thinking, competent models transmit knowledge and teach observers effective skills and strategies for managing environmental demands (Bandura et. al., 1994). Bandura also states that persuasive boosts in perceived self-efficacy lead people to try hard enough to succeed; these boosts promote the development of skills and a sense of personal efficacy. The hypothesis of this paper predicted a negative correlation between self-efficacy and peer pressure, backed by numerous studies including (Binnaz, 2012), where the negative correlation was significant, and (KiranEsen, 2003) who also found negative relationships between peer pressure and success at school. Ryan’s (2000) findings in a study with adolescents also support their results. Despite the hypothesis, results show a positive correlation between both variables. Taking into consideration the that the largest percentage of participating are STEMers - high achieving high school students who represent the vast majority of the sample- we can relate the direct relation to their natural tendency to prefer competitive environments, since they are used to it since almost kindergarten. When splitting results by gender responses, for females the relation is significantly weak. A proposed explanation to that is the nature of society in Egypt where males are generally more encouraged, hence the higher efficacy explained. As proposed in the hypothesis, there is a positive correlation between self-awareness and self-efficacy. When people have a transparent look deep down into their minds and souls, they are less prone to being affected by peers or society. This can be explained by the already present vivid self-image for highly aware people; they analyze their feelings and actions independently, without the need of a peer’s opinion or judgement. Given the limited number of participants in the study, the results represent a thin slice of Egypt’s adolescent community. Also, the high female to male ratio might have been the reason behind some differences in gender-split analysis. It is recommended to replicate the study over a greater sample and with close numbers of participants of both genders. Regarding the methods of sharing this study, participants filled out internet questionnaires with no monetary interest. There was no supervision and no motive but the good of their hearts. This could have impacted the time they devoted to completing the questionnaires or the authenticity of their answers. We had to shorten the original questionnaires to minimize the risk of randomizing answers, resulting in a potential drop in the accuracy of the scale. Despite the inevitable limitations, this study sheds light on the psychological welfare of the most disturbing human age, adolescence. As implied by the results, peer pressure is slightly correlated to self-efficacy. Taking these results to the field and working on creating a healthy environment where peer pressure can coexist with psychological prosperity in peace will have positive impacts on education quality. Motivation techniques like honoring ranking students or giving away symbolic gifts to the most hard-working students can be the base of building psychological core strength for teenagers. Because of the direct relation between self-awareness and self-efficacy, it is advised to offer self-awareness guides and workshops to students and high schoolers. Promoting self-awareness will have various extending impacts on different aspects of life, including less stress and lower risk of depression. Given the heavy reliance of psychological analysis on the context of test, it would contribute greatly to the accuracy of the study if the questionnaire was filled out in different periods. In this study for instance, participants filled the questionnaire during their final examinations. During exams, stress levels are abnormally high; this might have affected their reported self-efficacy. Though peers have the most significant role in each other’s lives during adolescence, they are not the only potential source of pressure. Community pressure and family pressure, external factors in this study, might be closely related to self-efficacy; the relation needs further investigation. For the sake of a better, healthier maturation of human beings, we need to work on the welfare of teenagers. Adolescence is a tough transition from pinkish easy life to tough responsibilities, work, and decision making. Awareness may not guarantee happiness; instead, it is a very strong protective shield that levels up the quality of life. This protective shield is strengthened by healthy relationships with peers. To invest in the quality-of-life of rising Homo sapiens is to ascend one more step towards a sustainable future.