Paper 2010/655 Original: eprint.iacr.org/2010/655 ↗

On the Affine Equivalence and Nonlinearity Preserving Bijective Mappings

Abstract

It is well-known that affine equivalence relations keep nonlineaerity invariant for all Boolean functions. The set of all Boolean functions, $\mathcal{F}_n$, over $\bbbf_2^n$, is naturally regarded as the $2^n$ dimensional vector space, $\bbbf_2^{2^n}$. Thus, while analyzing the transformations acting on $\mathcal{F}_n$, $S_{2^{2^n}}$, the group of all bijective mappings, defined from $\bbbf_2^{2^n}$ onto itself should be considered. As it is shown in \cite{ser,ser:dog,ser:dog:2}, there exist non-affine bijective transformations that preserve nonlinearity. In this paper, first, we prove that the group of affine equivalence relations is isomorphic to the automorphism group of Sylvester Hadamard matrices. Then, we show that new nonlinearity preserving non-affine bijective mappings also exist. Moreover, we propose that the automorphism group of nonlinearity classes, should be studied as a subgroup of $S_{2^{2^n}}$, since it contains transformations which are not affine equivalence relations.