Application of Quadratic Cryptanalysis for a Five Round XOR Modification of the Encryption Algorithm Gost 28147-89

In the paper, we give analysis for the mathematical foundations of quadratic approximations; singularities of a binary operation 〈∙, ∙〉_k in the vector field Z_2^m, mathematical foundations of quadratic approximations of a special form (Boolean functions), and quadratic approximations of block ciphers. We also present the mathematical foundations of constructing correlation matrices using quadratic approximations for estimating nonlinear reflections of S-blocks in the block coding algorithm GOST 28147-89. Examples of four-bit permutations recommended for use in S-blocks of the algorithm GOST 28147-89 [6] are considered. It is shown that in almost all cases there are more probable (compared to linear) quadratic relations of a special form on the input and output bits of these permutations. Quadratic approximations are developed for round transformations of the algorithm GOST 28147-89. Also, quadratic relations of a special form are developed for input and output bits for a five round XOR modification of the encryption algorithm GOST 28147-89. The solutions of the problems of applying quadratic cryptanalysis to find some bits of round keys are given.