Abstract
The Walsh transform \(\widehat {Q}\) of a quadratic function \(Q:\mathbb {F}_{p^{n}}\rightarrow \mathbb {F}_{p}\) satisfies \(|\widehat {Q}| \in \{0,p^{\frac {n+s}{2}}\}\) for an integer 0 ≤ s ≤ n−1. We study quadratic functions given in trace form \(Q(x) = {{\text {Tr}_{\mathrm {n}}}}({\sum }_{i=0}^{k}a_{i}x^{p^{i}+1})\) with the restriction that \(a_{i} \in \mathbb {F}_{p},~ 0\leq i\leq k\). We determine the expected value for the parameter s for such quadratic functions from \(\mathbb {F}_{p^{n}}\) to \(\mathbb {F}_{p}\), for many classes of integers n. Our exact formulas confirm that on average the value of s is small, and hence the average nonlinearity of this class of quadratic functions is high when p = 2. We heavily use methods, recently developed by Meidl, Topuzoğlu and Meidl, Roy, Topuzoğlu in order to construct/enumerate such functions with prescribed s. In the first part of this paper we describe these methods in detail and summarize the counting results.
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Acknowledgments
We thank an anonymous referee, whose comments improved the presentation of this paper.
The second author is supported by the Austrian Science Fund (FWF) Project no. M 1767-N26.
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Kaşıkcı, C., Meidl, W. & Topuzoğlu, A. Spectra of a class of quadratic functions: Average behaviour and counting functions. Cryptogr. Commun. 8, 191–214 (2016). https://doi.org/10.1007/s12095-015-0142-9
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DOI: https://doi.org/10.1007/s12095-015-0142-9