2019/06/02 by Semaev, Igor
#94A60 #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.1906.00426
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 (Nf,Hf) 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≥ 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≥ 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≥ 2. The definitions may be easily extended to q-ary functions.