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 $r\geq 2$ 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.
Metadata
- Available format(s)
-
PDF
- Category
- Secret-key cryptography
- Publication info
- Preprint. MINOR revision.
- Keywords
- Boolean functionsnonlinearityoptimal functionsbent functions
- Contact author(s)
- igor @ ii uib no
- History
- 2019-06-03: received
- Short URL
- https://ia.cr/2019/633
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2019/633, author = {Igor Semaev}, title = {New non-linearity parameters of Boolean functions}, howpublished = {Cryptology {ePrint} Archive, Paper 2019/633}, year = {2019}, url = {https://eprint.iacr.org/2019/633} }