A Method of Deterministically Generating Block Substitution Tables which Meet a Given Standard of Nonlinearity

Abstract

There is considerable interest in the cryptographic community in block substitution tables or S-boxes, which are highly nonlinear in some sense. This is particularly important in Feistel type systems of which DES is a prime example. In such systems, the key is used to interact with the clear text data and the substitution tables serve as barriers to limit access to the key by comparing clear text with cipher text data. The primary tools of cryptanalysis against Feistel type systems are differential and linear cryptanalysis. The principal foil against these is nonlinearity as typically measured by L ₁ and L ₄ norms using the Walsh-Fourier transform. These highly nonlinear tables are generally found by searching and their properties are determined empirically by testing rather than relying on underlying mathematical theory. The purpose of this paper is to find a method of deterministically generating block substitution tables of maximal nonlinearity, as measured by some recognized criteria without sacrificing other desirable cryptographic qualities.