Construction of resilient S‐boxes with higher‐dimensional vectorial outputs and strictly almost optimal non‐linearity

Abstract

Resilient substitution boxes (S-boxes) with high nonlinearity are important cryptographic primitives in the design of certain encryption algorithms. There are several trade-offs between the most important cryptographic parameters and their simultaneous optimization is regarded as a difficult task. In this paper we provide a construction technique to obtain resilient S-boxes with so-called strictly almost optimal (SAO) nonlinearity for a larger number of output bits m than previously known. This is the first time that the nonlinearity bound 2 n−1 − 2 n/2 of resilient (n, m) S-boxes, where n and m denote the number of the input and output bits respectively, has been exceeded for m > n 4. Thus, resilient S-boxes with extremely high nonlinearity and a larger output space compared to other design methods have been obtained.

On Resilient Boolean and Vectorial Boolean Functions with High Nonlinearity

“…By Construction 2, a 1-resilient (32, 4) functions with nonlinearity 32, 4, 1, 2 31 − 2 15 − 2 11 can be obtained. The nonlinearity of this function is better than the results in [19,20].…”

“…In Construction 3, the numbers of rows of the matrixs A and B should be enough to ensure that both ϕ i and ψ i are bijective mappings. Hence, the following inequation must hold N(u, m, t + 1) • (2 m − 1) • 2 u + M(k, m, t + 1) • 2 k ≥ 2 n , where the value of N(u, m, t + 1) and M(k, m, t + 1) can be found in [19,20].…”

“…How to construct resilient (n, m) functions with nonlinearity lower than the upper bound but larger than 2 n−1 − 2 n/2 is also an important problem in recent years. By contrast, fewer results have been obtained than the Boolean ones [17][18][19][20][21].…”

On cryptographic properties of (n + 1)-bit S-boxes constructed by known n-bit S-boxes

“…These functions reached the Siegenthaler's bound, and can be of optimal algebraic immunity or suboptimal algebraic immunity. In 2016, a construction of resilient S-boxes with higher-dimensional vectorial outputs and strictly almost optimal non-linearity was presented in [15]. A construction of highly nonlinear (n, m, t, d) resilient S-boxes with given algebraic degree was gave in [6].…”