In this paper we investigate the relationship between the nonlinearity and the order of resiliency of a Boolean function. We first prove a sharper version of McEliece theorem for Reed-Muller codes as applied to resilient functions, which also generalizes the well known Xiao-Massey characterization. As a consequence, a nontrivial upper bound on the nonlinearity of resilient functions is obtained. This result coupled with Siegenthaler’s inequality leads to the notion of best possible tradeoff among the parameters: number of variables, order of resiliency, nonlinearity and algebraic degree.We further show that functions achieving the best possible trade-off can be constructed by the Maiorana-McFarland like technique. Also we provide constructions of some previously unknown functions.
CITATION STYLE
Sarkar, P., & Maitra, S. (2000). Nonlinearity bounds and constructions of resilient Boolean functions. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1880, pp. 515–532). Springer Verlag. https://doi.org/10.1007/3-540-44598-6_32
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