Extending Nyberg construction on Galois fields of odd characteristic

Abstract

As is known, the Nyberg design S-boxes possess the cryptographic properties valuable for practical application. Up to date this construction has been considered only for fields of characteristic 2. This paper presents an extension of the Nyberg construction to the fields of odd characteristic. The notion of nonlinearity distance of p-function is introduced, and the affine ternary code is built. The Nyberg design S-boxes with fields characteristic p = 3 for all lengths N <= 243 are built. The nonlinearity distances are calculated, and it is shown that with an increase of S-box length, these distances increase essentially faster as compared to the fields of characteristic p = 2.