On the nonlinearity of idempotent quadratic functions and the weight distribution of subcodes of Reed-Muller codes

Abstract : The Walsh transform \hat{Q} of a quadratic function Q : F2^n → F2 satisfies |\hat{Q(b)}| ∈ {0, 2 n+s 2 } for all b ∈ F_{2^n} , where 0 ≤ s ≤ n − 1 is an integer depending on Q. In this article, we investigate two classes of such quadratic Boolean functions which attracted a lot of research interest. For arbitrary integers n we determine the distribution of the parameter s for both of the classes, C1 = {Q(x) = Tr_n(\sum^{(n−1)/2}_{ i=1} a_ix^{2^i +1}) : a_i ∈ F2}, and the larger class C2, defined for even n as C2 = {Q(x) = Tr_n(^{(n/2)−1}_ { i=1} a_ix^{2^i +1}) + Tr_n/2 (a_{n/2} x^{2^n/2 +1}) : a_i ∈ F2}. Our results have two main consequences. We obtain the distribution of the non-linearity for the rotation symmetric quadratic Boolean functions, which have been attracting considerable attention recently. We also present the complete weight distribution of the corresponding subcodes of the second order Reed-Muller codes.
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Pascale Charpin, Nicolas Sendrier, Jean-Pierre Tillich. The 9th International Workshop on Coding and Cryptography 2015 WCC2015, Apr 2015, Paris, France. 2015, Proceedings of the 9th International Workshop on Coding and Cryptography 2015 WCC2015. 〈wcc2015.inria.fr〉
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Nurdagül Anbar, Wilfried Meidl, Alev Topuzo˘ Glu. On the nonlinearity of idempotent quadratic functions and the weight distribution of subcodes of Reed-Muller codes. Pascale Charpin, Nicolas Sendrier, Jean-Pierre Tillich. The 9th International Workshop on Coding and Cryptography 2015 WCC2015, Apr 2015, Paris, France. 2015, Proceedings of the 9th International Workshop on Coding and Cryptography 2015 WCC2015. 〈wcc2015.inria.fr〉. 〈hal-01276432〉

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