Abstract:
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.