Constructions of Resilient S-Boxes With Strictly Almost Optimal Nonlinearity Through Disjoint Linear Codes

In this paper, a novel approach of finding disjoint linear codes is presented. The cardinality of a set of [u, m, t+1] disjoint linear codes largely exceeds all the previous best known methods used for the same purpose. Using such sets of disjoint linear codes, not necessarily of the same length, we have been able to provide a construction technique of t-resilient S-boxes F:F2ⁿ→2^m ( n even, ) with strictly almost optimal nonlinearity . This is the first time that the bound 2^n-1-2^n/2 has been exceeded by multiple output resilient functions. Actually, the nonlinearity of our functions is in many cases equal to the best known nonlinearity of balanced Boolean functions. A large class of previously unknown cryptographic resilient S-boxes is obtained, and several improvements of the original approach are proposed. Some other relevant cryptographic properties are also briefly discussed. It is shown that these functions may reach Siegenthaler´s bound n-t-1, and can be either of optimal algebraic immunity or of slightly suboptimal algebraic immunity, which was confirmed by simulations.