STUDY OF THE CRYPTOGRAPHIC STRENGTH OF THE S-BOX ObTAINED ON THE bASIS OF EXPONENTIATION MODULO

. This article presents one of the main transformations of symmetric block ciphers used to protect confidential information, a new method for obtaining a non-linear S block, and an analysis of the results obtained. The S-box obtained by this method can be used as a non-linear transformation in block cipher algorithms to protect confidential data transmitted over an open channel. In most well-known works in the field of analysis and synthesis of modern block symmetric ciphers, S-box is used as a mathematical apparatus for cryptographic Boolean functions. In this case, each S-box is represented by a set of composite Boolean functions whose properties characterize the efficiency of the nonlinear substitution node. Substitution nodes for modern symmetric primitives, including key unfolding functions, are usually implemented as replacement tables. Considering that in most modern block symmetric ciphers for introducing round keys, the encryption algorithm uses a linear operation (bitwise addition modulo 2), S-blocks are the only elements responsible for the cryptographic stability of block encryption algorithms. The required number of rounds of block symmetric ciphers is selected taking into account the results of the cryptographic analysis performed, provided that the properties of S-boxes are specified. As the main criteria and performance indicators, the balance and nonlinearity of composite Boolean functions are used; strict avalanche criterion (SAC), propagation criterion; algebraic degree; the value of the autocorrelation function. In this article, a study was made of the nonlinearity and strict avalanche criterion (SAC) of the S-box used in the block symmetric encryption algorithm. The results of the study were compared with the S-boxes of modern cryptographic algorithms and showed good results.


Introduction
The developed S-box must meet the following criteria: • S-block should increase the cryptographic strength of the symmetric block cipher and resistance to cryptanalysis; • S-block should be easy to create; • The S-block must meet the requirements of SAC, BIC, NL, LP, DP, etc. Within the framework of the project "Development of a means of cryptographic information protection for secure negotiations over HF radio'' of the Ministry of Education and Science of Kazakhstan, the AL03 encryption algorithm was developed [13]. Currently, the analysis of the cryptographic strength of the AL03 algorithm is being carried out. In this regard, a comprehensive study of the S-box, which is one of the transformations of this algorithm, is considered one of the most important tasks. The method for obtaining the S-box used in the AL03 encryption algorithm is described in [14].

Method for obtaining the S-box
Let's consider a way to obtain an S-box consisting of three steps. At the first, an irreducible polynomial is chosen that step generates a multiplicative group in the Galois field GF (2 8 ) and an irreducible polynomial is called a base [14]. The selected polynomial is exponentiated modulo P(x): . Then Sbox i is added modulo 2(XOR) with a fixed vector B(x): (2) In the third step, we multiply by the matrix (3) Table 1 shows the values obtained after converting the binary result obtained by formula (3) to the hexadecimal number system.

Algebraic analysis and simulation result
To assess the quality of the developed S-box, we used the standard algebraic analysis. This analysis includes the bit independence test, nonlinearity, strict avalanche test, and the probability of linear and differential approximation. The proposed S-box was also compared with some classical S-boxes and currently constructed S-boxes [7]. The proposed S-box corresponds to all the optimal values of the standard algebraic analysis. The details of this analysis are shown below.

Nonlinearity
According to [14], the degree of nonlinearity of Boolean functions NL(f) is defined as the minimum Hamming distance between the i-th Boolean function and linear functions.  Table 2 shows the comparative characteristics of the S-boxes of block symmetric ciphers AES, SM4, Ref [4], Ref [15], and the proposed S-box.
The graphical representation is shown in Figure 2. The nonlinearity of the proposed S-box is 100 (see Table 2). This value is higher than Ref [4] and Ref [15].
The Strict Avalanche Criterion (SAC) determines how much the output bits change when the input bits change once. An S-box satisfies the SAC criterion if a single bit change in the input bit causes about half of the output bits to change [7]. A comparison of the overall SAC analysis of the proposed S-box with AES [17], S-p-box, and Ref [15] is shown in Tables 3-6, while the mean results are shown in Table 7, and a graphical representation of the comparative analysis is shown in Figure 2. From Table 7 it can be seen that the proposed S-box has a maximum value of 0.526, a minimum value of 0.437, a mean value of 0.487, and a standard deviation of 0.015. 87

Conclusion
S boxes play an important role in ensuring the robustness of symmetric transformations. In order to protect the algorithms against various cryptanalysis methods, S-blocks must have a number of cryptographic properties and satisfy a number of criteria. In this paper, it has been shown that the nonlinear and strong avalanche effect (SAC) of the S-box used in the AL03 encryption algorithm has very good performance. The results of comparing the studied S-box with the S-boxes of known cryptographic algorithms were also presented. As a result of the comparison, the corresponding result was obtained with the popular S-boxes. The S-boxes that I compared in the article are dynamic, so they only use it once. And since our proposed S-box is stable, it can be used for any block encryption algorithm. The proposed S-box has all the necessary cryptographic properties, and it has been proven that it can be used in any cryptosystem. In further works, the cryptographic strength of S-box will be studied by other methods, and the results will be obtained. The fully studied S-box is used as the main nonlinear transformation method responsible for the crypto-strength of the block encryption algorithm developed according to the research conducted under the grant project.