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

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

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 keyRSA, 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

numbersWhere |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 1a2 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 VectorsSimplify the vectors such that |0> = i and |1> = j where

i=<1,0> and j=<0,1> are unit vectors in two spaceApplying the rotational matrix, M, to these vectors we get

that for any general , θ |x> = M* |0> = <cos( ), sin( )> and θ θ |y> = M*|1> = <-sin( ), cos( )>θ 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( ) + y*sin( )θ θ b = -x*sin( ) + y*cos( )θ θ

a and b are the coordinates of the vector <x,y> in basis{ |x> , |y> }

AssumptionsOne: A photon can be transported through optical fiber

without changing its polarizationTwo: 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 messageBob then measures the photons at t0

Required Properties of Theta-FunctionsAll 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

The second criteria is easily visible by the construction of f(t).

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: [email protected] • Phone: 903-490-2982• Institution: Southern Arkansas University