The substitution and permutation functions are the main functions of information cryptographic systems that provide diffusion and mixing of information. It is required to analyze these cryptographic primitives while creating new algorithms for data transformation using such functions. A new scale of notation has been proposed for performing such analysis. This is the factorial sets statistical series notation scale. This scale of notation helps to index the factorial sets statistical series

more »

... ents and establish a one-to-one correspondence between the number and a specific type of substitution. The use of this system provides a way to find new characteristics and properties of substitutions. Since substitutions of the factorial sets form a cyclic group, the operation on elements of this group can be determined. In the general case, the operation used is a multiplication of substitutions, but this paper suggests another operation for group determination. The article presents a capability for dividing the factorial sets substitutions. Two methods for implementing the substitution division operations are proposed such as a method of successive transition and a method of group transition, which are implemented through non commutative operation of multiplying on inverted substitutions.