Subgroups of symmetric groups of substitutions of a series of factorial sets

Abstract

Cryptographic algorithms and protocols are based on the simplest functions – permutations and substitutions of the elements of a given finite set. By combining and transforming these elements, we can form a structural element of information security, namely a mathematical transformation with a required degree of cryptographic strength. The main problem related to the development of such transformations is that when designing more powerful algorithms, developers use an increasing amount of data and computing power, while relying on the hardware capabilities of the information system. This increases resource consumption and limits the speed and confidentiality of applications. However, studying the mathematical aspects of the generation of permutations and the notation of series of factorial sets, we also need to investigate the problem of data accumulation in order to develop new algorithms that would reduce the models of hardware storage of a large number of permutations without limiting their capabilities. The notation of series of factorial sets allows us to use the algorithm for the formation of any element of the factorial set without storing the permutations in the RAM. Thus, it is no longer necessary for cryptographic algorithms to store data in arrays, since this function can be replaced by the algorithm for forming permutations using the notation of a series of factorial sets. The article considers axioms and methods of creating subgroups of symmetric substitution groups of a series of factorial sets. New concepts of a series of factorial sets and symmetric groups of substitutions of a series of factorial sets introduced in 2014, make it possible to expand the possibilities for the analyses of symmetric groups of substitutions. Namely, it is possible to number, identify, and structure these groups, and make the processes of group and individual transformation more visual. The article presents a classification of subgroups according to the formation method.