Paper 2019/633

New non-linearity parameters of Boolean functions

Igor Semaev

Abstract

The study of non-linearity (linearity) of Boolean function was initiated by Rothaus in 1976. The classical non-linearity of a Boolean function is the minimum Hamming distance of its truth table to that of affine functions. In this note we introduce new "multidimensional" non-linearity parameters $(N_f,H_f)$ for conventional and vectorial Boolean functions $f$ with $m$ coordinates in $n$ variables. The classical non-linearity may be treated as a 1-dimensional parameter in the new definition. $r$-dimensional parameters for $r\ge 2$ are relevant to possible multidimensional extensions of the Fast Correlation Attack in stream ciphers and Linear Cryptanalysis in block ciphers. Besides we introduce a notion of optimal vectorial Boolean functions relevant to the new parameters. For $$ and even $$ optimal Boolean functions are exactly perfect nonlinear functions (generalizations of Rothaus' bent functions) defined by Nyberg in 1991. By a computer search we find that this property holds for $$ too. That is an open problem for larger $$ and $$. The definitions may be easily extended to $$-ary functions.