=Paper=
{{Paper
|id=Vol-3137/paper12
|storemode=property
|title=The Pseudorandom Key Sequences Generator Based on IV-Sets of Quaternary Bent-Sequences
|pdfUrl=https://ceur-ws.org/Vol-3137/paper12.pdf
|volume=Vol-3137
|authors=Elena Bakunina,Artem Sokolov
|dblpUrl=https://dblp.org/rec/conf/cmis/BakuninaS22
}}
==The Pseudorandom Key Sequences Generator Based on IV-Sets of Quaternary Bent-Sequences==
https://ceur-ws.org/Vol-3137/paper12.pdf
The Pseudorandom Key Sequences Generator Based on IV‐Sets
of Quaternary Bent‐Sequences
Elena Bakunina a and Artem Sokolov b1
a
National University “Odessa Law Academy”, Fontanska road, 23, Odessa, 65000, Ukraine
b
Odessa Polytechnic National University, Shevchenko Avenue, 1, Odessa, 65044, Ukraine
Abstract
The development of modern cryptography, steganography, and communication systems with
noise-like signals often requires the use of non-binary generators of pseudorandom key
sequences. The main requirements for such a cryptographic construct are the compliance of
gamma generated by them with the criteria of NIST stochastic tests and the implementation
of the concepts of diffusion and confusion with respect to the generated gamma and the
cryptographic key. In this paper, we empirically confirm the prospects of combining the
advantages of quaternary bent-sequences, which are characterized by the highest nonlinearity
value, with the advantages of the linear feedback shift registers, which are characterized by a
high stochastic quality of the generated sequences. We present the scheme of a
pseudorandom key sequences generator based on binary linear feedback shift registers and
the IV-sets of quaternary bent-sequences. It has been established that the presented scheme
generates pseudorandom key sequences corresponding to all the NIST stochastic tests, while
the maximum values of the nonlinearity of quaternary bent-sequences provide a high level of
implementation of the concept of confusion. At the same time, construction of IV-sets of
quaternary bent-sequences was found, which provides the best stochastic properties of the
generated gamma. The developed generator is characterized by high values of the number of
protection levels and the simplicity of the algorithmic implementation. Compared to
combining two binary generators of pseudorandom key sequences based on dual sets of
binary bent-sequences, 40% fewer linear feedback shift registers are required to ensure the
operation of the developed generator. The practical application of the developed generator is
justified in systems that require many-valued logic pseudorandom key sequences.
Keywords
Pseudorandom key sequences generator, bent-sequence, many-valued logic.
1. Introduction and statement of the problem
The Pseudorandom Key Sequences Generator (PRKSG) is a very important construct used in
modern cryptographic applications as well as in other areas of science and technology: PRKSG is the
basis for the functioning of modern stream encryption algorithms, organizing modern efficient modes
for block encryption algorithms [1], defining the steganographic path in modern steganographic
algorithms, functioning of modern radio communication systems based on Frequency-Hopping
Spread Spectrum (FHSS) technology [2], etc.
Today, there are many different layouts for the PRKSG construction: the classical scheme based
on Linear Feedback Shift Register (LFSR) with the use of nonlinear elements [3, 4], schemes based
on the theory of dynamic chaos [5], cellular automata [6], etc. Regardless of the chosen scheme, the
following basic requirements must be applied to PRKSG being developed today: they must be
characterized by the high level of stochastic quality (the level of which is measured by the compliance
with a generally accepted set of NIST stochastic tests [7]), the ability to generate pseudorandom key
CMIS-2022: The Fifth International Workshop on Computer Modeling and Intelligent Systems, Zaporizhzhia, Ukraine, May 12, 2022
EMAIL: elenabakunina72@gmail.com (E. Bakunina); radiosquid@gmail.com (A. Sokolov)
ORCID: 0000-0002-5700-7321 (E. Bakunina); 0000-0003-0283-7229 (A. Sokolov)
© 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org) Proceedings
sequences based on a short cryptographic key (at the same time, the PRKSG must provide a high level
of implementation of Shannon’s concepts of diffusion and confusion [8] between the key element and
the generated gamma), the simplicity of the algorithmic implementation, and the high performance of
the PRKSG algorithm.
The classical LFSR-based PRKSG layout with a nonlinear element fully fulfills the mentioned
requirements, in particular, its modification based on bent-sequences proposed in [9].
The development of the theory of quantum cryptography, as well as the recently proposed
cryptographic algorithms based on many-valued logic functions [10], have led to the need to create
PRKSG operating on a non-binary alphabet A {0,1,..., q 1}, q 2 . Thus, the scheme of the PRKSG
based on LFSR and dual sets of bent-sequences was generalized to operate over the alphabet
A {0,1, 2} , which corresponds to the problems of quantum cryptography.
However, the vast majority of information today is stored, processed, and transmitted in binary
form, which makes it especially relevant to consider the possibility of constructing PRKSG that
would operate over the alphabets A {0,1,...,2 k 1} , primarily over the quaternary alphabet
A {0,1,2,3} . The use of the quaternary alphabet fundamentally makes it possible to simplify the
PRKSG schemes by generating twice more gamma bits during one cycle, and also increases the
possibility of implementing the principles of diffusion and confusion in the PRKSG [8]. The
quaternary alphabet also provides a much greater variety of algebraic constructions that can be used to
build high-quality PRKSG.
Steganography is another important application of quaternary PRKSG in view of the peculiarities
of the choice of the steganographic path: the ability to select one of the three color components or skip
a given pixel from the process of information embedding.
These facts substantiate the tasks of further research on the possibilities of improvement of the
classical layout for constructing the PRKSG operating over the quaternary alphabet.
The purpose of this paper is to develop a quaternary PRKSG based on IV-sets of bent-sequences.
2. The theoretical foundations for the proposed PRKSG
The classic layout of the PRKSG implies the use of LFSR as the main component.
Definition 1 [11]. An LFSR is a shift register consisting of d memory cells, the value of the input
element of which is determined by the value of the function constructed in accordance with the
primitive irreducible over the Galois field GF (2) polynomial of degree d .
It is known [10] that the use of LFSR makes it possible to achieve good diffusion, and also
provides a sufficiently high stochastic quality of the generated pseudorandom key sequences, which
will be shown below in Table 2 on the example of LFSR built on the basis of the primitive irreducible
polynomial
f ( x) x83 x 46 x 45 x 1 . (1)
The disadvantages of LFSR include the fact that they generate pseudorandom key sequences in
accordance with a fairly simple rule defined by a primitive irreducible polynomial.
This does not allow them to provide a sufficiently high level of nonlinearity of the relationship
between the elements of the short key and the generated pseudorandom key sequence. In other words,
these constructions do not provide a sufficient level of confusion. This circumstance leads to the
possibility of launching attacks against such a generator, for example, using the Berlekamp-Massey
algorithm.
One of the historical attempts to overcome this shortcoming is the Geffe Generator, which
provides a significantly higher level of confusion compared to the direct application of the LFSR
through the use of several interconnected LFSR units. However, this level of confusion is also
insufficient due to the simplicity of the links between the pooled LFSR units.
In modern PRKSG, this drawback is eliminated by using a non-linear element in conjunction with
LFSR, which is often represented by the Boolean function with a high level of nonlinearity. Thus, the
block diagram of PRKSG in its classical layout is shown in Fig. 1.
Fig. 1: Block diagram of PRKSG based on LFSR and nonlinear element
The ideal option for nonlinear element is to use such special algebraic constructions as bent-
functions (whose truth tables are called as bent-sequences), which have the maximum level of
nonlinearity, and, accordingly, provide the highest level of confusion implemented by PRKSG.
In accordance with the definition [9], a binary sequence B [b0 , b1 ,..., bi ,..., bN 1 ] , where bi {1}
are coefficients, of even length N 22 m , i 0,1,..., N 1 is called a bent-sequence if it has uniform
absolute values of its Walsh-Hadamard spectrum WB ( ) , which can be represented in matrix form
WB ( ) BA , 0,1,..., N 1 , (2)
where A is the Walsh-Hadamard matrix of order N .
The Walsh-Hadamard matrices are constructed in accordance with the Sylvester construction,
which is specified using the following recurrent rule
A k 1 A2 k 1 (3)
A2 k 2 , A1 1 .
A2 k 1 A2 k 1
Despite the successful layout of the block diagram shown in Fig. 1, the practical application of
bent-sequences in it is extremely difficult due to bent-sequences imbalance, as a result of which the
gamma generated by PRKSG is also unbalanced, and, therefore, does not meet the most basic criteria
of stochastic quality.
Another undoubted disadvantage of using binary bent-sequences is their existence only for lengths
N 2 2 m 4,16,64,256,... , while binary bent-functions of an odd number of variables do not exist,
which limits the scalability of PRKSG schemes.
In [9], for the binary case, a solution to this problem was proposed based on the use of dual sets of
maximally nonlinear bent-functions.
This PRKSG scheme showed excellent characteristics both in terms of performance and in terms
of the quality of the generated pseudorandom key sequences, which is confirmed by both a set of
stochastic tests [12], as shown in [9], and a set of NIST stochastic tests [7]. The number of protection
levels of this scheme reaches the value 2,67 10 49 2165 , which exceeds the number of protection
levels of the AES-128 block symmetric cipher [13].
This scheme was also modified for the ternary case in [14] using LFSR based on primitive
irreducible polynomials over the Galois field GF (3) , as well as triple sets of bent-sequences. Despite
the absence of specific NIST tests for ternary pseudorandom sequences, the sequences generated by
PRKSG [13] fully correspond to the set of tests [12], which suggests their high level of quality.
To construct a quaternary PRKSG, it is proposed to use the complete class of quaternary bent-
sequences described in [15] and synthesized in [16]. Let us introduce the definition of the Vilenkin-
Chrestenson transform that we will need, as well as the definition of the many-valued logic bent-
sequence.
Definition 2 [16]. The coefficients of the Vilenkin-Chrestenson transform of a q-valued logic
function is the vector obtained by multiplying its truth table T (represented in the exponential form)
by the complex conjugate of the Vilenkin-Chrestenson matrix
A T V 16 . (4)
while for the case of 4-functions the Vilenkin-Chrestenson matrix is constructed according to the
following recurrent rule
Vk Vk Vk Vk
V V 1 V 2 V 3
V4 k 1 ,
k k k k
(5)
Vk Vk 2 Vk Vk 2
Vk Vk 3 Vk 2 Vk 1
where “+” is the operation of addition mod4, the matrices V are represented in symbolic form, i.e. the
summation is performed with respect to the indices zi , and
z 0 e e j0 e j0 e j0
j0
z0 z0 z0
z 3
z1 z 2 z3 e j 0 e 2 e j e 2
j j
V4 0 . (6)
z 2 z 0 z 2 e j 0 e j e j 0 e j
z0
3
z3 z 2 z1 e j 0 e j 2 e j e j 2
z0
Definition 3 [16]. For the Vilenkin-Chrestenson matrix of order N q k , a bent-sequence
j 2
H [h0 , h1 ,..., hi ,..., hN 1 ] is a sequence over the alphabet hi e q , 0,1,..., q 1 if it has a
uniform absolute values of the Vilenkin-Chrestenson spectrum, which can be represented in matrix
form
H ( ) H V N const , 0,1,..., N 1 , (7)
where VN is the Vilenkin-Chrestenson matrix of order N over the alphabet
j 2
hi e q , 0,1,..., q 1 .
Bent-sequences are very important mathematical objects for cryptographic constructions
development, however, their main drawback, which limits their use in cryptography, in particular, in
the problems of development of cryptographically secure PRKSG, is their imbalance, the inequality
of the number of symbols 0,1,..., q 1 contained in them, i.e. K 0 K 1 ... K q 1 . This drawback was
solved in [9] and [14] by using dual sets and triple sets of bent-sequences respectively, while the
definition of dual sets and triple sets was generalized in [16], resulting in the definition of a q-set of
bent-sequences which was presented.
Definition 4 [16]. A set of q q-ary bent-sequences is called a q-set if the concatenation of their
truth tables is balanced, i.e. it satisfies the relation K 0 K 1 ... K q 1 .
The research of the complete class of quaternary bent-sequences of length N 16 makes it
possible to distinguish the following weight structures, which are presented in Table 1 in the
following form {K 0 , K 1 , K 2 , K 3} .
Table 1
Weight structures of quaternary bent‐sequences of length N 16
Weight Weight Weight Weight
J J J J
structure structure structure structure
{10,0,6,0} 192 {4,6,0,6} 2112 {8,2,4,2} 6336 {3,3,7,3} 10240
{0,10,0,6} 192 {6,4,6,0} 2112 {2,8,2,4} 6336 {3,3,3,7} 10240
{6,0,10,0} 192 {1,5,5,5} 6144 {4,2,8,2} 6336 {2,4,6,4} 25152
{0,6,0,10} 192 {5,1,5,5} 6144 {2,4,2,8} 6336 {4,2,4,6} 25152
{0,6,4,6} 2112 {5,5,1,5} 6144 {7,3,3,3} 10240 {6,4,2,4} 25152
{6,0,6,4} 2112 {5,5,5,1} 6144 {3,7,3,3} 10240 {4,6,4,2} 25152
Based on the data presented in Table 1, as well as on the Definition 4, it is fundamentally possible
to construct 3948 IV-sets, which can be formed on the basis of a complete class of quaternary bent-
sequences of length N 16 . As the experiments show, the best stochastic quality of the generated
pseudorandom key sequences is provided with the following choice of their structure
[ H ( H 1) mod 4 ( H 2) mod 4 ( H 3) mod 4] , (8)
where H is one of the quaternary bent-sequences of length N 16 , the cardinality of the complete
class of which is J 200704 .
Note that theoretically, as a LFSR, it is possible to use registers based on primitive irreducible
polynomials over the extended field GF (2 2 ) . A method for synthesizing such a primitive irreducible
polynomials over extended fields is presented in [17]. Based on this method, for example, we can
construct following polynomial
f ( x) x10 x 9 3x 8 3x 7 2 x 6 x 4 3x 3 x 2 2 , (9)
on the basis of which the LFSR scheme shown in Fig. 2 can be built.
Fig. 2: LFSR scheme based on primitive irreducible polynomial (9)
Note that the operations of multiplication and summation in the LFSR scheme shown in Fig. 2
are performed in the extended Galois field GF (2 2 ) , which arithmetic is determined by a single
irreducible polynomial over the Galois field GF (2) of second degree f ( x) x 2 x 1 in accordance
with the following tables
0 1 2 3 0 1 2 3
0 0 1 2 3 0 0 0 0 0
1 1 0 3 2, 1 0 1 2 3 . (10)
2 2 3 0 1 2 0 2 3 1
3 3 2 1 0 3 0 3 1 2
However, the results of the performed research which are presented in Table 2 show that
LFSR constructed using extended fields show significantly worse stochastic performance compared to
LFSR constructed using prime Galois fields. This circumstance does not allow them to be applied in
the developed quaternary PRKSG. In our opinion, a good solution is to use binary LFSR with their
subsequent multiplexing to ensure that arguments are supplied to the input of the IV-set of bent-
sequences, as well as the choice of a specific quaternary bent-sequence from the IV-set is performed.
3. PRKSG scheme and its characteristics
Based on the above theoretical information, we present a quaternary PRKSG scheme based on
a IV-set of quaternary bent-sequences. The following primitive irreducible polynomials are used for
constructing LFSR (in accordance with the results of [9], in order to maximize the period of the
generated pseudorandom key sequences, the degrees of primitive irreducible polynomials are chosen
as coprime)
LFSR1 : f1 ( x) x 23 x17 x11 x 5 1;
LFSR 2 : f 2 ( x) x 29 x 20 x16 x11 x8 x 4 x 3 x 2 1;
LFSR 3 : f 3 ( x) x 31 x 21 x12 x 3 x 2 x 1;
(11)
LFSR 4 : f 4 ( x) x19 x 9 x8 x 5 1;
LFSR 5 : f 5 ( x) x17 x16 x 3 x 1;
LFSR 6 : f 6 ( x) x 37 x 28 x18 x 9 1,
and also, the IV-set of quaternary bent-sequences is chosen in accordance with condition (8)
H1 {0331132333230113};
H 2 {1002203000301220};
(12)
H 3 {2113310111012331};
H 4 {3220021222123002}.
In Fig. 3, we present the scheme of the developed PRKSG based on the IV-set of quaternary bent-
sequences (12) and LFSR constructed in accordance with primitive irreducible polynomials (11).
Fig. 3: Scheme of the proposed PRKSG based on IV‐sets of bent‐sequences (12)
Note that in view of the formation of a IV-set of quaternary bent-sequences in accordance with
rule (8), in contrast to the PRKSG schemes presented in [9] and [14], there is no need to store the
entire IV-set, as well as implement a selection block, which can be replaced by a summation device,
which greatly simplifies the technical implementation of PRKSG and saves required for its
implementation memory cells.
Thus, the values of quaternary bent-functions H 2 , H 3 , H 4 can be formed by adding a constant 1, 2
or 3 to the original bent-function H 1 , respectively. At the same time, the simultaneous operation of
LFSR 5 and LFSR 6 allows choosing these values with equal probabilities, which leads to ensuring
the equiprobable choice of one of the quaternary bent-sequences from the IV-set at each iteration of
the PRKSG operation.
Let us note the features of the operation of the proposed PRKSG scheme based on IV-sets of
quaternary bent-sequences. Binary registers LFSR1 and LFSR 2 generate pseudorandom sequences,
which are subsequently multiplexed into quaternary sequences that determine the first input variable
of the bent-function (equivalent to corresponding bent-sequence). Similarly, the binary registers
LFSR 2 and LFSR 3 define the second input variable of the quaternary bent-function (equivalent to
corresponding bent-sequence).
The registers LFSR 5 and LFSR 6 after the multiplexer generate one of the values from the set
{0,1,2,3} , which determines the choice (with help of summation) of quaternary bent-sequence from
the IV-set. Next, the corresponding value of the quaternary bent-function (equivalent to corresponding
bent-sequence) is calculated, which is the output value of the PRKSG at a current cycle of operation.
The number of protection levels of the developed PRKSG is determined both by the number of
possible initial states of the LFSR and by the number of bent-sequences in the complete class. In our
case, when using primitive irreducible polynomials (11), as well as the complete class of quaternary
bent-sequences [16], the number of protection levels of the developed PRKSG is defined as
(223 1)(229 1)(231 1)(219 1)(217 1)(237 1) 200704 2173,6 , (13)
which exceeds the number of protection levels of the AES-128 block symmetric cipher.
At the same time, we note that in order to obtain a quaternary PRKSG based on a combination of
two binary PRKSG units based on dual sets of bent-sequences [9], we would have to use 10 LFSR
items, which is 40% more than in the proposed scheme. Thus, the use of IV-sets of quaternary bent-
sequences makes it possible to simplify the algorithmic implementation of PRKSG in the applications
where quaternary pseudorandom key sequences are required.
We note an important property of the proposed PRKSG scheme: it is easily scalable and makes it
possible, if necessary, to easily increase the number of protection levels by using quaternary IV-sets
of bent-sequences of greater length.
For example, consider the possibility of using quaternary bent-sequences of length N 64 . We
take as a basis one of the quaternary bent-sequences of given length N 64
H1 [3333301203213131000001231032020233333012032131312222230132102020] , (14)
on the basis of which, considering the construction (8), we obtain the IV-set
H1 {3333301203213131000001231032020233333012032131312222230132102020};
H 2 {0000012310320202111112302103131300000123103202023333301203213131};
(15)
H 3 {1111123021031313222223013210202011111230210313130000012310320202};
H 4 {2222230132102020333330120321313122222301321020201111123021031313}.
In view of the fact that each of the bent-sequences (15) is the equivalent of the corresponding bent-
function of three variables, to construct the corresponding PRKSG we need another two LFSR to
ensure the presence of an input signal for each of the variable of equivalent bent-function, as well as
two LFSR for ensuring the selection of the bent-sequence from the IV-set.
Thus, in addition to the primitive irreducible polynomials (11), we choose two additional primitive
irreducible polynomials
LFSR 7 : f 7 ( x) x 41 x 3 1;
(16)
LFSR 8 : f 8 ( x) x 43 x 6 x 4 x 3 1.
On Fig. 4 we present a PRKSG scheme operating using eight LFSR and a IV-set of quaternary
bent-sequences (15) of length N 64 .
Fig. 4: Scheme of the proposed PRKSG based on IV‐sets of bent‐sequences (15)
The number of protection levels of the developed PRKSG is determined by the number of possible
initial states of the eight LFSR included in it. In the case of using primitive irreducible polynomials
(11) and (16), we obtain the following value
(2 23 1)(2 29 1)(231 1)(219 1)(217 1)(237 1)
(17)
(2 41 1)(2 43 1)(2 47 1) 2 287.
The obtained value of the number of protection levels exceeds the value of the number of
protection levels of the AES-256 cryptographic algorithm.
Note that, to date, there are practically no methods for synthesizing complete classes of quaternary
bent-sequences of length N 16 represented in the literature, which leads to the decrease in the total
value of protection levels number for developed PRKSG shown in Fig. 4. This circumstance naturally
poses the practical problem of developing such a methods in order to increase the number of
protection levels for PRKSG based on IV-sets of quaternary bent-sequences.
In Table 2, we present the results of the NIST stochastic tests of the developed PRKSG, its
structural elements as well as some known analogues.
Table 2
Results of the NIST stochastic tests
No. Test LFSR based LFSR based PRKSG Developed Developed
on (1) on (9) [10] PRKSG PRKSG
(Fig. 3) (Fig. 4)
P- Pass P- Pass P- Pass P- Pass P- Pass
value rate value rate value rate value rate value rate
1 Monobit test 0.51 ✔ 0.86 ✔ 0.30 ✔ 0.09 ✔ 0.18 ✔
2 Frequency within
0.74 ✔ 0.73 ✔ 0.99 ✔ 0.47 ✔ 0.06 ✔
block test
3 Runs test 0.72 ✔ 0.85 ✔ 0.11 ✔ 0.40 ✔ 0.72 ✔
4 Longest run ones
0.14 ✔ 0.80 ✔ 0.04 ✔ 0.75 ✔ 0.19 ✔
in a block test
5 Binary matrix rank
0.76 ✔ 0 ✘ 0.08 ✔ 0.87 ✔ 0.78 ✔
test
6 DFT test 0.84 ✔ 0 ✘ 0.79 ✔ 0.34 ✔ 0.68 ✔
7 Non overlapping
template matching 0.99 ✔ 0.97 ✔ 1 ✔ 1 ✔ 1 ✔
test
8 Overlapping
template matching 0.01 ✔ 0.98 ✔ 0.26 ✔ 0.89 ✔ 0.08 ✔
test
9 Maurers universal
0.8 ✔ 0.01 ✔ 0.16 ✔ 0.40 ✔ 0.23 ✔
test
10 Linear complexity
0.16 ✔ 0 ✘ 0.83 ✔ 0.24 ✔ 0.93 ✔
test
11 Serial test 0.9 ✔ 0.99 ✔ 0.18 ✔ 0.04 ✔ 0.81 ✔
12 Approximate
0.9 ✔ 0.99 ✔ 0.18 ✔ 0.04 ✔ 0.81 ✔
entropy test
13 Cumulative sums
0.49 ✔ 0.99 ✔ 0.18 ✔ 0.06 ✔ 0.16 ✔
test
14 Random excursion
0.33 ✔ 0.04 ✔ 0.16 ✔ 0.17 ✔ 0.04 ✔
test
15 Random excursion
0.42 ✔ 0 ✘ 0.29 ✔ 0.07 ✔ 0.03 ✔
variant test
The analysis of the data presented in Table 2 leads to the conclusion that the proposed PRKSG
based on IV-sets of quaternary bent-sequences of length N 16 , as well as quaternary bent-sequences
of length N 64 has a high level of stochastic quality and fully complies with all the NIST stochastic
tests.
The NIST test results presented in Table 2, as well as the obvious simplicity of the technical
implementation of the proposed PRKSG structure, allow us to recommend to use both developed
PRKSG variants (Fig. 3 and Fig. 4) for practical use.
4. Conclusion
We note the main results of the research:
1. It has been established that the construction of quaternary PRKSG is possible on the basis of
binary LFSR, as well as IV-sets of quaternary bent-sequences. At the same time, the configuration of
the IV-set was found, which provides the best stochastic properties of the gamma generated by the
PRKSG.
2. The scheme of the PRKSG is proposed which is based on binary LFSR and IV-set of quaternary
bent-sequences. The proposed PRKSG provides a high level of stochastic properties of the generated
sequences and the number of protection levels that can be easily scaled. For the case of use of the
complete class of quaternary bent-sequences of length N 16 , the number of protection levels
reaches the value 2 209, 65 , while in the case of using bent-sequences of length N 64 , the number
of protection levels reaches the value 2 287 , which exceeds the number of protection levels of the
AES-256 cryptographic algorithm. Developed PRKSG also requires 40% smaller number of LFSR
for generation of quaternary pseudorandom key sequences compared to combining of two PRKSG,
based on dual sets of bent-sequences.
3. The presented PRKSG is of interest from the point of view of practical application where
quaternary pseudorandom key sequences are needed, for example stream encryption algorithms
operating based on many-valued logic principles, communication systems based on the FHSS
technology, steganographic algorithms with the possibility of pseudorandom selection of a
steganographic path.
Considering the above material, an important direction for further research is the development of
methods for the synthesis of complete classes of quaternary bent-sequences, which will expand the
variety of possible structures of the developed PRKSG, as well as increase the number of its
protection levels value.
Another possible direction for further development of the proposed PRKSG is its modification in
order to use larger values of base q.
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