On the construction of s-boxes of 4 X 4 size

The paper proposes and implements a method for constructing all 4x4 size s-boxes, satisfying a number of cryptographic properties: 1) bijectivity; 2) the absence of fixed points; 3) the nonlinearity of all non-trivial linear combinations of the coordinate functions; 4) the value of the difference characteristic p_s of the substitution s is not more than 4/16. A list of all 4x4 size s-boxes (a total of 383 084 314) with these properties is obtained. The created software is used to determine the specified characteristics of DES algorithm s-boxes (32 s-boxes of 4x4 size using fixations of bits controlled by one of four substitutions of degree 16 of s-box) and eight s-boxes recommended for use in the GOST 28147-89 algorithm. According to the DES algorithm, it is established that 6 s-boxes do not provide the property 2, 16 s-boxes have p_s = 8/16, 14 s-boxes have p_s = 6/16, and there is one s-box with p_s = 4/16 and one s-box with p_s = 10/16. According to the GOST 28147-89 algorithm, it is established that 3 s-boxes do not provide the property 2, all s-boxes have p_s = 4/16.

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