SYNTHESIS METHOD OF TERNARY BENT-FUNCTIONS OF THREE VARIABLES

Context. Such perfect algebraic constructions of many-valued logic as ternary bent-functions and their truth tables which are called as 3-bent-sequences, are used very often in modern cryptographic algorithms, in particular, in pseudorandom sequence generators. However, today there are no methods for synthesizing the ternary bent-functions class for a number of variables greater than two, which significantly limits the ability to scale the number of protection levels of the of pseudorandom sequence generators based on the ternary bent-functions. This circumstance generates the task of developing methods for the synthesis of ternary bent-functions, which is solved in this paper for the case of ternary bent-functions of three variables. The object of this research is the process of efficiency increasing of the cryptographic algorithms based on the functions of many-valued logic. Objective. The purpose of the paper is to construct a method for the synthesis of the set of ternary bent-functions of three variables. Method. The mathematical apparatus of the Reed-Muller transform (algebraic normal form) was used as the basis of the proposed constructive method for the synthesis of ternary bent-functions of three variables. So, on the basis of the established properties of the algebraic normal form of ternary bent-functions and limited enumeration, the search for ternary bent-functions up to affine terms is performed, after which we apply the procedure of reproduction. Results. As a result of using of the proposed method for the synthesis of ternary bent-functions of three variables, 155844 3-bent-sequences were found up to an affine term, while the cardinality of the full set of found 3-bent-sequences is 12623364. The research performed made it possible to determine that in this set there are 3-bent-sequences of six different weight structures, on the basis of which 12 different triple sets can be compiled for use in pseudorandom sequence generators. A scheme for a cryptographically stable pseudorandom sequence generator based on the found set of 3-bent-sequences of length 27 N  is proposed. It is shown that the protection levels number of such a generator of pseudorandom sequences is 41 7.041 10    which is comparable with the protection levels number of modern block symmetric cryptographic algorithms, for example, AES-128. Conclusions. The further development of modern cryptographic algorithms, in particular, cryptographically stable pseudorandom sequence generators, is largely based on the use of perfect algebraic constructions of many-valued logic. For the first time, a constructive method for the synthesis of ternary bent-functions of three variables is proposed. For the found set of ternary bent-functions, the distribution of weight structures is found, and the possible triple sets are established. Based on the constructed set of ternary bent-functions, a pseudorandom sequence generator scheme is proposed that has a protection levels number that is comparable with modern block symmetric cryptographic algorithms. We note that the constructed class of ternary bent-functions can also be used for the synthesis of cryptographically strong S-boxes, codes of constant amplitude, as well as error correction codes. As an actual area of further research, we can note the development of methods for the synthesis of ternary bent-functions of a larger number of variables.


ABBREVIATIONS
PRSG is a pseudorandom sequence generator; LFSR is a linear feedback shift register; FCSR is a feedback with carry shift register; BF is a ternary bent-function; ANF is an algebraic normal form.

NOMENCLATURE
N is a length of 3-bent-sequence; , , K K K is a amounts of symbols "0", "1" and "2" correspondingly;  is a polynomial of ANF;

INTRODUCTION
The current stage of development of cryptographic methods of information protection is characterized by the introduction of cryptographically high-quality functions of many-valued logic [1].At the same time, one of the most actual tasks is the development of ternary pseudorandom sequence generators (PRSG).Such generators are used in the tasks of quantum cryptography, and can also be used in implementations of cryptographic algorithms based on many-valued logic functions on binary computers.
At present, the dynamically developing and applicable in practice is the construction of ternary PRSG [2], which is based on the use of LFSR or FCSR [3] and a special nonlinear element, which is most often used with perfect algebraic constructions such as 3-bent-functions.At the same time, the number of protection levels for the PRSG depends on the length of the 3-bent-function and on the cardinality of the set of available 3-bent-functions, which makes it necessary to develop methods for synthesizing large sets of ternary bent-functions.
At present, regular methods have been created for the synthesis of 3-bent-functions of two variables, nevertheless, there are no such methods for a larger number of variables, which significantly reduces the possibility of increasing the number of protection levels of PRSG based on ternary bent-functions.
The object of research is the process of improving the efficiency of cryptographic algorithms based on many-valued logic functions.
The subject of research is the methods for the synthesis of 3-bent-functions.
The purpose of the work is to construct a method for the synthesis of the set of ternary bent-functions of three variables.The problem is to find in this set of sequences of length 27 N  such a sequences that are truth tables of ternary bent-functions of three variables, i.e. possess a uniform absolute values of Vilenkin-Chrestenson transformants.

REVIEW OF THE LITERATURE
The current stage in the development of information technology is characterized by the widespread introduction of the mathematical apparatus of many-valued logic functions into correcting coding [4] and information compression algorithms [5], as well as in the signal processing [6].
The current stage is also characterized by a rapid development of methods of many-valued logic functions and their implementation in cryptography.In particular, the use of cryptographically strong ternary PRSG, which are proposed to be used to increase the cryptographic strength of quantum information protection protocols, is proposed in [7,8].The authors of [9] proposed a scheme of effective PRSG based on ternary bent-functions presented in Fig. 1.

Figure 1 -Scheme of PRSG based on ternary bentfunctions
The raw material for the operation of the scheme (Fig. 1) are such perfect algebraic constructions as ternary bent-functions that have a uniform absolute value of Vilenkin-Chrestenson transformants and, accordingly, the maximum possible value of nonlinearity.
The Vilenkin-Chrestenson spectrum of the discrete sequence above the alphabet    is found by multiplying the column vector containing the samples of the signal by the complex conjugate transformation matrix V [10].
In this case, the matrix of the Vilenkin-Chrestenson transform of order 3 L , L   is constructed over the alphabet {0,1, 2} using the recurrence formula [11], and then in order to perform the Vilenkin-Chrestenson transform it is translated into the exponential form, i.e. to the : A generalized definition of a ternary bent-function was given in [11], according to which the existence of bent-functions of the many-valued logic of an odd number of variables was confirmed.Definition 1.For a Vilenkin-Chrestenson matrix of order k N q  , where q is a prime, a bent-sequence   is a sequence over an alphabet 2π ν , ν 0,1,..., 1 if it has a uniform absolute values of Vilenkin-Chrestenson spectrum that can be represented in matrix form ( ) where N V is the Vilenkin-Chrestenson matrix of order N over the alphabet 2π ν , ν 0,1,..., 1 Currently, methods for the synthesis of 3-bentsequences of two variables are known [12].The research of the structure of this class of bent-sequences of cardinality 486 J  allowed to establish their possible weight structures, depending on which they are classified into 6 classes: where the numbers in curly brackets show, respectively, the number of characters "0", "1" and "2" in the 3-bentsequence, and the numbers in parentheses indicate the number of 3-bent-sequences with the indicated structure.
In [9], it was established that, from the point of view of constructing an PRSG, the triple sets of 3-bentsequences possess the best properties.Definition 2. A set of three bent-sequences 1 2 3 , , B B B in the Vilenkin-Chrestenson basis is called a triple set if the concatenation of their truth tables in symbolic form is balanced, i.e. the number of characters "0" is equal to the number of characters "1" and is equal to the number of characters "2", i.e. 0 1 2 . Thus, the 3-bent-sequences from (4) determine two triple sets Currently, in the literature there are no methods for the synthesis of 3-bent-sequences of three variables, and accordingly, their weight structures remain unknown, which makes it impossible to construct specific PRSG schemes based on 3-bent-sequences of length 27 N  .

MATERIALS AND METHODS
The use of the exhaustive method for the synthesis of 3-bent-sequences, as it was done in [9], is ineffective for 3-bent-sequences of length 27 N  , since the solution of this problem will be coupled with enumeration of the set of all ternary sequences of length 27 N  , which consists of 27 3 7 625597 484987 full J   elements.At the same time, practice shows that the construction of purely regular methods for the synthesis of 3-bent-sequences is also difficult due to the complexity and unpredictability of this class of perfect algebraic constructions.Nevertheless, the performed researches show that the synthesis of 3-bent-sequences of length 27 N  can be performed in the Reed-Muller transformants domain, i.e. in algebraic normal form (ANF) [13].
Definition 3.An algebraic normal form of a q- function is a polynomial  over q Z of a degree deg( ) q   with coefficients {0,1,..., 1} i a q   , containing the operations "Sum modulo q" and "Multiplication modulo q".On the basis of Definition 3 the definition of affine functions, which play a key role in cryptography is introduced.
Definition 4. Ternary functions whose ANF polynomial has degree deg( ) 1   are called as affine.
Thus, each ternary sequence Moreover, it was established in [3] that the transition to the Reed-Muller transformants domain and vice versa can be performed in matrix form: where For example, let us consider some ternary sequence for which, taking into account (6), using the Reed-Muller transform matrix 27 L , we find the Reed-Muller transform coefficients vector [000000001000011120121120100] A  .(10) Each element of the ANF vector (10) is one corresponding coefficient in the general form of the ANF 3function of three variables ( , , ) x x x a a x a x a x a x x a x x a x a x x a x x a x a x x a x x a x x a x x x a x x x a x x a x x x a x x x a x a x x Thus, taking into account (11), we can write the ANF of sequence (9): Substituting consistently all possible sets of 1 2 3 { , , } x x x in (12), we obtain the original truth table (9).
Let us check if sequence ( 9) is a 3-bent-sequence.For this, it is necessary to find the absolute values of its Vilenkin-Chrestenson transform coefficients  Thus, sequence ( 9) is indeed a 3-bent-sequence.

EXPERIMENTS
Let us consider each ANF coefficient of the ternary sequence i a , where i is the vector of their three coordinates, each of which indicates the degree of occurrence of the corresponding variable in the term.Let us denote by ( ) i wt T i   the algebraic degree of the corresponding term.
As the algebraic degree of nonlinearity we call the maximum value among all degrees of terms included in the algebraic normal form of a ternary sequence The proposed constructive synthesis method for 3bent-sequences is based on the use of the Reed-Muller transform domain and the following determined properties of ternary bent-functions.
Property 1.The sum of a 3-bent-sequence with an affine sequence is a 3-bent-sequence.
Property 2. The maximum algebraic degree of the term included in the 3-bent-sequence of length 27 Thus, each element of the sequence { } i A a  , 0,1,..., 26 i  corresponds to the ANF term of 3-function in general form, as shown in the Table 1.In the Table 1 affine terms are marked by the symbol " α ", while terms whose algebraic degree exceeds the value ( ) 4 i wt T  are marked by the symbol "x".The remaining terms are marked by the symbol "*".
Analysis of the data presented in Table 1 shows that the number of terms marked by the symbol "*" is 19.Thus, we write the proposed constructive method for the synthesis of 3-bent-sequences of length 27 N  in the form of following specific steps.
Step x x x * ---Substituting the next specific values over the alphabet {0,1, 2} instead of the symbols "*", and substituting instead of the symbols " α " and "x" the values 0, we obtain a specific ternary sequence in the Reed-Muller transformants domain.This sequence, by multiplying by the matrix of the inverse Reed-Muller transform 1 27 L  , is transferred to the time domain, obtaining the candidate sequence F .
Step 2. We find the absolute values of the spectral coefficients of the Vilenkin-Chrestenson transform of the sequence F and check it for the compliance with the conditions of Definition 1.If it is a 3-bent-sequence, we save the corresponding sequence A , otherwise we discard it.
Step 3. If the end of the search is not reached among all possible sequences A , we go to Step 1, otherwise the search is completed.
Step 4. For each 3-bent-sequence found, we release 4 positions marked by the symbol " α ", i.e. the positions corresponding to affine terms.Substituting into them all possible values from the alphabet {0,1, 2} we obtain, on the basis of each 3-bent-sequence, a set of 4  3 81  3-bent-sequences.
Note that since in Table 1 there are 19 values marked with the symbol "*", the total number of ternary sequences that must be enumerated in the proposed algorithm is

RESULTS
Performing Steps 1,...,3 of the developed algorithm allowed us to find 155844 3-bent-sequences of length 27 N  up to affine terms.As an example, we can give one of the found 3-bentsequences in the form of its ANF as well as a sequence in time domain Note that among the found 3-bent-sequences there are 468 3-bent-sequences with an algebraic degree deg{ } 2 B  , 3744 3-bent-sequences with an algebraic degree deg{ } 3 B  and 151632 3-bent-sequences with an algebraic degree deg{ } 4 B  .Accordingly, based on the obtained set of 3-bentsequences, by applying Step 4 of the proposed algorithm, it is possible to synthesize a set of 3-bent-sequences with cardinality allowed us to establish that they can be classified according to the 6 possible weight structure types where, similar to (4), the numbers in curly brackets show, respectively, the number of characters "0", "1" and "2" in the 3-bent-sequence, and the numbers in parentheses indicate the number of 3-bent-sequences with the indicated structure.

DISCUSSION
The obtained set of 3-bent-sequences of length 27 N  is valuable not only from the point of view of the theory of synthesis of perfect algebraic constructions, but also can serve as a basis for constructing effective PRSG according to a scheme similar to Fig. 1.In view of the use of 3-bent-sequences of greater length, even the use of one possible triple set can provide a significant level of cryptographic stability.The PRSG scheme based on 3-bentsequences of length 27 N  is shown on Fig. 2. Let us determine the number of protection levels of the developed PRSG.For example, let the polynomials of the following mutually simple degrees to be chosen to construct the corresponding 3LFSR The number of primitive irreducible polynomials of degree and of degree The initial state of 3LFSR 1 can be selected by which is a significant value, and exceeds the number of protection levels of such a modern block symmetric cryptographic algorithm as AES-128 [15] and many other modern block symmetric chippers [16].

CONCLUSIONS
The scientific novelty lies in the fact that a method for the synthesis of 3-bent-sequences in the Reed-Muller transformants domain was developed.Using the developed method, a class of 3-bent-sequences of length 27 N  and cardinality 12 623364 J  was constructed.For the found class of 3-bent-sequences, six possible weight structures and 12 possible triple sets were discovered.
The practical significance consists in the fact that the synthesized class of 3-bent-sequences of length 27 N  was proposed to be used to construct the scheme of the ternary cryptographically strong PRSG.At the same time, the estimated number of protection levels of the constructed PRSG is 41 7.041 10    , which is comparable with the number of protection levels of modern block symmetric cryptographic algorithms.
As a further area of research, it is worth to note the development of methods for the synthesis of 3-bentsequences of longer lengths, as well as research of perfect algebraic constructions with larger values of q .

F
of complete enumeration, which is in 6561 times less.
the found class of 3-bent-sequences of length 27 N 

Figure 2 -
Figure 2 -PRSG scheme based on 3-bent-sequences of length 27 N  from one triple set 3-bent-sequences themselves in the triple set can be selected by 3 2103894 ways.Thus, the number of protection levels of the constructed PRSG is defined 1 kL  is the matrix of the inverse Reed-Muller transform, which is constructed in accordance with[13]; k L is the Reed-Muller transform matrix.