On a heuristic approach to constructing bijective vector Boolean functions with given cryptographic properties

Bijective vector Boolean functions (permutations) are used as nonlinear primitives of many symmetric ciphers. In this paper, we study a generalized construction of $(2m,2m)$-functions using monomial and arbitrary $m$-bit permutations as constituent elements. A heuristic algorithm for obtaining bijective Boolean functions with given nonlinearity and differential uniformity, based on this construction, is proposed. For this, a search is carried out for auxiliary permutations of a lower dimension using the ideas of spectral-linear and spectral-difference methods. The proposed algorithm consists of iterative multiplication of the initial randomly generated $4$-bit permutations by transposition, selecting the best ones in nonlinearity, the differential uniformity, and the corresponding values in the linear and differential spectra among the obtained $8$-bit permutations. The possibility of optimizing the calculation of cryptographic properties at each iteration of the algorithm is investigated; $8$-bit $6$-uniform permutations with nonlinearity $108$ are experimentally obtained.