Quadratic functions with prescribed spectra (original) (raw)

Abstract

We study a class of quadratic _p_-ary functions \({{\mathcal{F}}_{p,n}}\) from \({\mathbb{F}_{p^n}}\) to \({\mathbb{F}_p, p \geq 2}\), which are well-known to have plateaued Walsh spectrum; i.e., for each \({b \in \mathbb{F}_{p^n}}\) the Walsh transform \({\hat{f}(b)}\) satisfies \({|\hat{f}(b)|^2 \in \{ 0, p^{(n+s)}\}}\) for some integer 0 ≤ s ≤ n − 1. For various types of integers n, we determine possible values of s, construct \({{\mathcal{F}}_{p,n}}\) with prescribed spectrum, and present enumeration results. Our work generalizes some of the earlier results, in characteristic two, of Khoo et. al. (Des Codes Cryptogr, 38, 279–295, 2006) and Charpin et al. (IEEE Trans Inf Theory 51, 4286–4298, 2005) on semi-bent functions, and of Fitzgerald (Finite Fields Appl 15, 69–81, 2009) on quadratic forms.

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Authors and Affiliations

  1. Sabancı University MDBF, Orhanlı, Tuzla, 34956, Istanbul, Turkey
    Wilfried Meidl & Alev Topuzoğlu

Authors

  1. Wilfried Meidl
  2. Alev Topuzoğlu

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Correspondence toAlev Topuzoğlu.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

This work was partially supported by TUBITAK project 111T234.

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Meidl, W., Topuzoğlu, A. Quadratic functions with prescribed spectra.Des. Codes Cryptogr. 66, 257–273 (2013). https://doi.org/10.1007/s10623-012-9690-6

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