Abstract
The nonlinearity of a Boolean function \(F: \mathbb{F}_{2}^{m}\rightarrow \mathbb{F}_{2}\) is the minimum Hamming distance between f and all affine functions. The nonlinearity of a S-box \(f: \mathbb{F}_{2}^{m}\rightarrow \mathbb{F}_{2}^{n}\) is the minimum nonlinearity of its component (Boolean) functions \(v\cdot f,\, v\in \mathbb{F}_{2}^{n}\,\backslash \{0\}\). This notion quantifies the level of resistance of the S-box to the linear attack. In this paper, the distribution of the nonlinearity of (m, n)-functions is investigated. When n = 1, it is known that asymptotically, almost all m-variable Boolean functions have high nonlinearities. We extend this result to (m, n)-functions.
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Dib, S. Asymptotic nonlinearity of vectorial Boolean functions. Cryptogr. Commun. 6, 103–115 (2014). https://doi.org/10.1007/s12095-013-0090-1
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DOI: https://doi.org/10.1007/s12095-013-0090-1