The influence of S-boxes on the arrival of the cipher to the state of random substitution

Abstract

The subject of this study is the process of developing block symmetric ciphers with improved cryptographic stability indicators for solving the problems of information protection in information and communication systems. The goal of this study is to develop a mathematical model of the optimal s-box design (non-linear substitution block) for modern block symmetric ciphers. Task: to show that the stability of block symmetric ciphers does not significantly depend on the indicators of the s-boxes included in these ciphers; to justify their use without reducing the stability of random substitutions as s-box constructions of modern block symmetric ciphers; and to prove the futility of searching for s-boxes with special properties. The methods used are: methods of probability theory; mathematical statistics; combinatorics and system analysis; methods of statistical tests; and methods of Boolean algebra. The following results were obtained. In modern ciphers, nonlinear substitution transformations are used as S-boxes (in most ciphers, byte substitutions are used). S-boxes perform the main function of encryption transformation - mixing of text bits. Dynamic indicators of the arrival of the cipher in the state of random substitution depend on their effectiveness. The presented results confirm that the indicators of cipher stability do not significantly depend on the indicators of the S-boxes included in the ciphers. It is not the individual properties of substitutions, but the results of their sequential interactions decisive for achieving stability indicators. It is shown that the fee for encryption is chosen to be three to four times greater than the depth of the avalanche effect - the minimum number of cycles of the cipher’s arrival to the state of random substitution) is hardly noticeable. Conclusions. The scientific novelty of the obtained results is as follows. This work represents a generalization of the transition to the use of random permutations as S-boxes is an increase of only one cycle in the number of cycles of the arrival of the cipher to the state of random permutations, and this is hardly noticeable against the background of the stability margin used in modern ciphers. The result of this generalization is a scientific statement that can be described as follows: random S-boxes can be used in all known modern ciphers without reducing stability. That is, the task of finding S-boxes with special properties loses any meaning, which means that the scientific direction related to the search for S-boxes with special properties becomes redundant.