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Use of Time as a Quantum Key

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- By Caleb Parks and Dr. Khalil Dajani

Use of Time as a Quantum Key

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

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

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

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.

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> }

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

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

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

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])

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

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.

Simulation with Bob

Simulation with Eve

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.

Contacting Me

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