**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 $r\geq 2$ 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.

**Category / Keywords: **secret-key cryptography / Boolean functions, nonlinearity, optimal functions, bent functions

**Date: **received 2 Jun 2019

**Contact author: **igor at ii uib no

**Available format(s): **PDF | BibTeX Citation

**Version: **20190603:072843 (All versions of this report)

**Short URL: **ia.cr/2019/633

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