=Paper=
{{Paper
|id=Vol-1900/paper29
|storemode=property
|title=Symmetric encryption algorithm using “twisted” light
|pdfUrl=https://ceur-ws.org/Vol-1900/paper29.pdf
|volume=Vol-1900
|authors=Sergey A. Burlov,Alexander V. Gorokhov
}}
==Symmetric encryption algorithm using “twisted” light ==
https://ceur-ws.org/Vol-1900/paper29.pdf
Symmetric encryption algorithm using “twisted” light
S. A. Burlov1 , A. V. Gorokhov1
1 Samara National Research University, 34, Moskovskoye shosse, Samara, 443086, Russia
Abstract
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.
Keywords: quantum cryptography, encryption algorithm, orbital angular momentum of photons, “twisted” light
1. Introduction
The modern bit cryptography is developing rapidly due to the active development of information storage and transmission
devices. The search for good algorithms among algebraic structures leads to the known problems of discrete logarithm and
factorization, which have a large history of applications in cryptanalysis.
The quantum cryptography was appeared, potentially having unlimited information capacity, huge transmission speed and
stability, based on the laws of quantum physics. Among the types of algorithms for quantum cryptography are known only a
schemes of key distribution and their using is considered mainly as an auxiliary position. The possibilities of quantum encryption
are not yet fully disclosed, but steps to this are done every day. At present widely used the quantum two-dimensional systems
based on the particles spin states and polarization states of photons. The main goal of this paper consist in the use for encryption
a potentially infinite-dimensional quantum systems based on the states of orbital angular momentum of photons [1].
2. Orbital angular momentum of photon
The light beams with an azimuthal phase that depends on a complex factor exp(−ilφ) carry an orbital angular momentum.
The angle φ is the azimuthal coordinate in the cross section of the beam, and l can be any integer number. The value of l indicates
the amount of twist of the spiral phase front. The value OAM is equal to L = l · ~ per photon [1].
Many researches of this phenomenon are connected with a certain type of light beam - the Laguerre-Gaussian mode. In
works [2, 3] showed the modification scheme of the quantum key distribution (QKD), the transmission of information with
superposition of states with non-zero OAM values of photons [4]. Many works are related to the generation of the light beams
with OAM [5, 6, 7, 8]. The main difficulty in the practical use of this phenomenon consist in the problem of measurement
the OAM value of a photon and in search of the appropriate transmission medium for such beams. Some methods have been
developed, which allows to measure OAM value of a photons: the measurement method with generating hologram [9], the sorting
method using the Mach-Zehnder interferometers cascade [10, 11, 12], the method of optical geometric transformation [13, 14]
and etc.
In this paper we offer to use a method that uses generating holograms and cascade of Mach-Zehnder interferometers. It is
proposed to construct a measurement scheme in a such way as to minimize the uncertainty of the receiving beam. The absence
of photons at the output of the detector is also an useful unambiguos information for the process.
It is well known that when studying the states of photons with an orbital angular momentum, we get to an infinite-dimensional
Hilbert space, which is formed from the set of eigenstates of the operator Lbz :
∂
Lbz = i (1)
∂φ
In principle, states with an arbitrary OAM value may be generated in an experiment. In the paper [15] it is shown the
possibility of continuous beam generation with various values of OAM using computer-controlled holograms.
3. Merkley’s scheme
The basis of algorithm of the Merkley’s scheme is a secret super-growing sequence of the natural numbers
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Computer Optics and Nanophotonics / S.A. Burlov, A.V. Gorokhov
X
j−1
A = {a1 , a2 , ..., ak }, where a j ≥ ai , (2)
i=1
which distributed between the subscribers of the network (Alice, Bob, · · · ) and pair of numbers n and w
n, w ∈ N, n > 2 · ak , GCD(n, w) = 1. (3)
Here GCD(n,w) means the greatest common divisor of the numbers n, w, and the number n is greater than the sum of elements
of the sequence (2) [16]. Next, the numbers n and w create the new sequence according to the rule:
G = {g1 , g2 , ..., gk }, where g j = a j · w(mod n). (4)
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..[ ]. (5)
k
After it the corresponding sum is calculated
X
k
ci = g j · mi j . (6)
j=1
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 (6).
fi = ci · w−1 (mod n) (7)
This number is decomposed on the sequence (2) basis and as result the original message is obtained. These actions are
performed for all blocks. The reliability and validity analysis of this scheme can be found, for example, in the article [17].
4. Adapted Merkley’s schemes
Let the secret sequence (2) and secret numbers (3) are distributed between subscribers of the network. The permutation
sequence T is formed by the sequence (4)
T = σ(G) = {g j1 , g j2 , ..., g jk }, where g j1 < g j2 < ... < g jk (8)
using the substitution !
1 2 ··· k
σ= . (9)
j1 j2 · · · 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:
Bi = σ(Mi ) = {bi1 , bi2 , ..., bik }. (10)
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 (7). The resulting number is decomposed based on the
sequence (2).
4.1. Variant I
Here, in order to encrypt the transmitted text, it is suggested to use a mixed state, which corresponds to superposition
X
k E
|Ψi = ai · bi · OAM = g ji , (11)
i=1
here factors ai are given by the sender and factors bi are calculated in accordance to the bit decomposition (10), and
fi = ci · w−1 (mod n). (12)
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Figure 1: Scheme of the process of formation, transmission and measurement of packages
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 (4).
· · · ··· ··· ··· ··· ··· · · ·
· · · α21 ··· 0 ··· 0 · · ·
· · · ··· ··· ··· ··· ··· · · ·
ρ = · · · 0 ··· α22 ··· 0 · · · . (13)
· · · ··· ··· ··· ··· ··· · · ·
· · ·
0 ··· 0 ··· α2k · · ·
··· ··· ··· ··· ··· ··· ···
The SLM control unit generates a mixed state (11) and transmits it during the iteration period. In this case, the input
measurement block should detect which states are participating in the generation of the mixed state. According to [10], the
cascade of the Mach-Zehnder interferometers can “do this work”. But there is one important feature: to determine 2 p states,
2 p − 1 interferometers required.
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 (6), then calculate the expression (7) and get the source text.
4.2. Variant II
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 B′i is obtained during the time interval τ = νk , 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.
X
k
cj = b′i · gi . (14)
i=1
Then, the expression (7) should be calculated and the resulting number is decomposed using a basis of the secret sequence (2). As
result, the opentext block is obtained. Having received all the blocks and deciphering them, the recipient decrypts the transmitted
message.
5. Conclusion
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
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138
Computer Optics and Nanophotonics / S.A. Burlov, A.V. Gorokhov
sequence: the probability of determining key is equal k!1 . 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 (2) and the maximum index of the
OAM of the beam, which will be detected with a minimum error. Based on the maximum ”well” detectable value f of the beam
orbital angular momentum, the length of the bit sequence can not exceed the value of log2 ( f ), whereas the maximum length is
reached for the ”bad” superincreasing sequence {1, 2, 4, 8, 16, 32, · · · }.
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).
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