Methods of linear and differential relations in cryptography

The paper studies mappings $F$ of binary vector spaces of large dimensions. The mappings are assumed to be represented by deep branching superpositions of local non-linear mappings of low-dimensional spaces. We propose and investigate the methods for construction of probabilistic linear and differential relations connecting the arguments and the values of a mapping $F$. Relation selection is based on optimization not the exact probability of satisfying these relations, but some approximation of it since it is easier to estimate. We prove theorems on exact values of the probability of satisfying the relations obtained, identify the shortcomings and features of the proposed approach to the relation construction and illustrate them by a number of examples. We discuss the role of the developed theory for cryptographic synthesis.