Use of Time as a Quantum Key

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

2. 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

3. 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

4. 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

5. Photon Polarization • The polarization of a photon can be expressed in bra-ket notation in terms of two state vectors |x> and |y> asa*|x> + b*|y> with a2+ b2 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 • a2 is the probability that the polarization is in the |x> state. • b2 is the probability that the polarization is in the |y> state.

6. Determination of the Basis Vectors • Simplify the vectors such that |0> = i and |1> = j where i=<1,0> and j=<0,1> are unit vectors in two space • Applying the rotational matrix, M, to these vectors we get that for any general θ, |x> = M* |0> = <cos(θ), sin(θ)> and |y> = M*|1> = <-sin(θ), cos(θ)> • 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 <x,y> in basis{ |x> ,|y> }

7. 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*t5+B*t4+Ct3 + Dt2 + 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

8. 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 t0 • Alice then sends this time t0 to Bob in a classically-encrypted message • Bob then measures the photons at t0

9. Required Properties of Theta-Functions • All the theta-functions intersect in exactly one point which will be called θ0 at time t0 • 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

10. Generation of the Functions • Set f(t) = A*t5+B*t4+Ct3 + Dt2 + Et + F • Property two can be determined as follow • Set f(t) - θ0= (t-t0)(t-ai)(t+ai)(t+bi)(t-bi) where i is the imaginary number, then f(t) - θ0 has only one zero which means f(t) = θ0 at only one point. • Expand the right side then, • At5+Bt4+Ct3+Dt2+Et+(F- θ0) = t5+(-t0)t4 + (a2+b2)t3 + (-t0[a2+b2])t2 +(a2b2)t + (-t0[a2b2])

11. Generation (Cont) • By identification of variables, A = 1; B = -t0; C = a2+b2; D = -t0*C; E = a2b2; F = θ0 – Et0 • C and E are free variables • f(t) = t5 - t0*t4+Ct3 - Ct0*t2 + Et + θ0 - Et0 • Finally, f(t0) = (t0)5 – t0*(t0)4+C(t0)3 – Ct0*(t0)2 + Et0 +θ- Et0 = θ0 • The second criteria is easily visible by the construction of f(t).

12. 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 t0 • 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.

13. Simulation with Bob

14. Simulation with Eve

15. 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.

16. Contacting Me • Name: Caleb Parks • Email: cparks1000000@gmail.com • Phone: 903-490-2982 • Institution: Southern Arkansas University