New non-linearity parameters of Boolean functions
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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≥ 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.
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