An algorithm for applying a ”twisted” light for constructing an encryption scheme is described. Our approach is founded on famous classical symmetric permutation algorithm based on NP-full task for ”Knapsack Problem” with changes taken into account the quantum origin of the information carrier. As a measuring device for selection of pure states from a mixed one, the Mach-Zehnder interferometer cascade is supposed to use, which allows sorting the parity of the mixed state of the orbital angular momentum (OAM) of photons.
The basis of algorithm of the Merkley’s scheme is a secret super-growing sequence of the natural numbers which distributed between the subscribers of the network (Alice, Bob, · · · ) and pair of numbers n and w
A = {a1, a2, ..., ak}, where a j ≥ j−1 X ai, i=1 n, w ∈ N, n > 2 · ak,
GCD(n, w) = 1.
k |Ψi = X ai · bi · OAM = g jiE ,
i=1
fi = ci · w−1(mod
n).
here factors ai are given by the sender and factors bi are calculated in accordance to the bit decomposition (
G = {g1, g2, ..., gk}, where g j = a j · w(mod n).
An original message is divided into blocks of bits of length k n M = {m1, m2, ... mn}, j ∈ 1..n ⇒ {Mi} = {mi1, mi2 ... mik}, i ∈ 1..[ ].
k After it the corresponding sum is calculated
k ci = X g j · mi j.
j=1 fi = ci · w−1(mod n)
This number is a block of encrypted text that is transmitted to another subscriber of a network. In its turn, the receiver
calculates the value fi from the obtained value ci given by expression (
This number is decomposed on the sequence (
using the substitution
Let the secret sequence (
T = σ(G) = {g j1, g j2, ..., g jk},
where g j1 < g j2 < ... < g jk σ = 1 j1 2 · · · j2 · · · k ! jk .
The control device for spatial light modulator (SLM) is being configured to generate laser beams with OAM photon projection only for values from the set T . The generation of target beams can be realized, for example, using computer-controlled holograms of diffraction gratings according to Refs [9].
The opentext is converted to a bit string. Each block is processed separately. The i-th iteration is performed as follows: Schematically, the design of the sender and receiver of the encryption process is shown in Fig. 1.
The digital-to-analog converter (DAC) specify SLM to generate the required beam type. Below are two versions of the
encrypted text generation that correspond to light rays with OAM of different types. The measurement block is also different
for each option, but the result of his work is the same: we get a list of values, which were laid in the ciphertext. This data is
transferred to the computer and deciphered by computing of expression (
Here, in order to encrypt the transmitted text, it is suggested to use a mixed state, which corresponds to superposition
Bi = σ(Mi) = {bi1, bi2, ..., bik}.
(
It is necessary to obtain a mixed state with the density matrix ρ. In general, the density matrix has k2 elements, but for a
mixed state only the diagonal elements can differ from zero, which correspond to the elements of the sequence (
· · · · · · · · · · · · · · · · · · · · · ρ = ····· ····· ····· ··α00··12·· ····· ····· ····· ··α00··22·· ····· ····· ····· ··α00··k2·· ····· ····· ····· .
· · · · · · · · · · · · · · · · · · · · ·
The SLM control unit generates a mixed state (
To optimize the measurement, it is proposed to use a short cascade. Optimization is based on the fact that for sorting out k values (knowing these values), each photon will pass no more than p interferometers. In total, we need a maximum of k · p interferometers. Therefore, when building a cascade, one can block empty paths, thereby greatly reducing the number of constituent elements.
Having received the statistics, one need to select those indicators that satisfy the specified threshold values, find their sum,
which corresponds to (
In this variant it is proposed to use a sequence of pure OAM photons states as an encryption text. The SLM control unit, before starting the transmission, gives the beamforming device a control signal for sending the zero Gaussian mode to the receiver. The receiver and the sender should be synchronized during the iteration period - ν of the beams sequence transmission. When the sequence Bi′ is obtained during the time interval τ = kν , depending on the value of 0 or 1, the SLM sends a pure state corresponding to gi j or its inversion.
Reception is carried out after receiving the signal state, which can be a zero Gaussian mode, and during a time interval νk the
detector perceives the beam with predetermined OAM value. If it is not detected, 0 is sent. Each iteration needs a time interval
equal ν. After the successful transfer of one packet the sum of indicators that are assumed to be equals to 1 is calculated.
(
i=1
Then, the expression (
Described encryption scheme is symmetric scheme due to restrictions imposed earlier, so that for effective measurement it is necessary to minimize the uncertainty of the received signal for legal subscribers. This can be done primarily due to the fact that the legal subscriber knows what sequence and what physical signals should be received and the messages themselves are unknown a priori.
The persistence of the presented variant I is determined by the durability of the classical Merkley’s scheme. The reliability of
the variant II schema is determined by a stability of the permutational interrelations, which are used to calculate the transmitted
sequence: the probability of determining key is equal k1! . Therefore the length of the original sequence should be optimal.
Optimum in this case is understood as a weighting between the length of the cipher sequence (
For an eavesdropper, obtaining a stream without accurate detection does not provide any information about the signal being transmitted, because the zeros of the sequence are sent also by a non-zero OAM value. Negative sign of the projection of the orbital angular momentum also needs to be revealed, for this the eavesdropper will be given a very short time interval (therefore it is important that the carrier can not be uniquely stored).