# New non-linearity parameters of Boolean functions

## TL;DR

This paper introduces multidimensional non-linearity parameters for Boolean functions, extending classical measures and exploring their implications for cryptanalysis and optimal function design.

## Contribution

It defines new multidimensional non-linearity parameters for Boolean functions and investigates their properties and relevance to cryptographic attacks and optimality.

## Key findings

- Classical non-linearity is a special case of the new parameters.
- Optimal vectorial Boolean functions correspond to perfect nonlinear functions for certain parameters.
- Computer search suggests the property extends to some higher dimensions, posing open problems.

## 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 $r\geq 2$ 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 $r=1$ and even $n\geq 2m$ 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 $r=2, m=1, n=4$ too. That is an open problem for larger $n,m$ and $r\geq 2$. The definitions may be easily extended to $q$-ary functions.