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Use of Time as a Quantum Key By Caleb Parks and Dr. Khalil Dajani

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What is Quantum Cryptography? In general, quantum computing involves using quantum particles such as electrons or photons in computations Cryptography involves sending sensitive information safely Quantum cryptography is simply cryptography using quantum methods Quantum cryptography is governed by the laws of quantum mechanics

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Why Do We Need Quantum Cryptography? Many classical algorithms already exist but a large number of them require a secret key RSA, one of the leading forms of encryption, relies on the difficulty of finding prime factors. Shor's algorithm can break RSA encryption with approximate speed of O((log N) 3 ) (where n is the number of bits in the key) Conclusion: RSA is not secure

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Definitions: A theta-function is a function which controls the angle of polarization of a photon A critical time is a time at which a number of theta- functions intersect. A photon is charged if it is governed by some theta- function

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Photon Polarization The polarization of a photon can be expressed in bra-ket notation in terms of two state vectors |x> and |y> as a*|x> + b*|y> with a 2 + b 2 equal 1, and a and b are complex numbers Where |x> and |y> form a basis for some Bloch Sphere (basically, the space where quantum states exist) One can assign |x> to 0 while |y> equals 1 a 2 is the probability that the polarization is in the |x> state. b 2 is the probability that the polarization is in the |y> state.

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Determination of the Basis Vectors Simplify the vectors such that |0> = i and |1> = j where i= and j= are unit vectors in two space Applying the rotational matrix, M, to these vectors we get that for any general θ, |x> = M* |0> = and |y> = M*|1> = The scalars for these vectors a, b such that a*|x> + b*|y> = V (where V is any vector on the Bloch Sphere) are as follows: a = x*cos(θ) + y*sin(θ) b = -x*sin(θ) + y*cos(θ) a and b are the coordinates of the vector in basis { |x>, |y> }

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Assumptions One: A photon can be transported through optical fiber without changing its polarization Two: There exists a mechanism to cause a photon's polarization to change as a specific function of time The function must be of the form f( A*t 5 +B*t 4 +Ct 3 + Dt 2 + Et + F ) where f is any function and A, B, C, D, E, and F are controlled by the mechanism. Three: There exists some way to maintain a photon's state for a period of time. Four: One can measure photons in an arbitrary basis

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The Algorithm in Brief Suppose Alice wants to send a message to Bob. Alice will then notify Bob that she wants to communicate. Alice then sends quantum bits charged so that the message appears at a critical time t 0 Alice then sends this time t 0 to Bob in a classically- encrypted message Bob then measures the photons at t 0

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Required Properties of Theta- Functions All the theta-functions intersect in exactly one point which will be called θ 0 at time t 0 The functions are all of odd degree of 5 or more. Thanks to the unsolvability of the quintic equation, no one will be able to determine the zero of the equations by a formula even if they can obtain the formula

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Generation of the Functions Set f(t) = A*t 5 +B*t 4 +Ct 3 + Dt 2 + Et + F Property two can be determined as follow Set f(t) - θ 0 = (t-t 0 )(t-ai)(t+ai)(t+bi)(t-bi) where i is the imaginary number, then f(t) - θ 0 has only one zero which means f(t) = θ 0 at only one point. Expand the right side then, At 5 +Bt 4 +Ct 3 +Dt 2 +Et+(F- θ 0 ) = t 5 +(-t 0 )t 4 + (a 2 +b 2 )t 3 + (-t 0 [a 2 +b 2 ])t 2 +(a 2 b 2 )t + (-t 0 [a 2 b 2 ])

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Generation (Cont) By identification of variables, A = 1; B = -t0; C = a 2 +b 2 ; D = -t 0 *C; E = a 2 b 2 ; F = θ 0 – Et 0 C and E are free variables f(t) = t 5 - t 0 *t 4 +Ct 3 - Ct 0 *t 2 + Et + θ 0 - Et 0 Finally, f(t 0 ) = (t 0 ) 5 – t 0 *(t 0 ) 4 +C(t 0 ) 3 – Ct 0 *(t 0 ) 2 + Et 0 + θ - Et 0 = θ 0 The second criteria is easily visible by the construction of f(t).

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Safety of the Algorithm Why is this more secure than the classical encryption which secured the agreed critical time? Alice will ensure that no one can break the code in a time less than t 0 By the time that any eavesdropper has determined the critical time, the information will already be gone. Multiple (at least ten) theta-functions will be used in the algorithm.

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Simulation with Bob

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Simulation with Eve

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Summary The algorithm takes little time compared to many quantum encryption algorithms No eavesdropper can gain information about the message Alice sends There is no need for a secret key to use this method The computations used in the algorithm are easy and efficient.

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Contacting Me Name: Caleb Parks Phone: Institution: Southern Arkansas University

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