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Spectra of a class of quadratic functions: Average behaviour and counting functions

Kaşıkçı, Canan and Meidl, Wilfried and Topuzoğlu, Alev Spectra of a class of quadratic functions: Average behaviour and counting functions. Cryptography and Communications . ISSN 1936-2447 (Print) 1936-2455 (Online) Published Online First http://dx.doi.org/10.1007/s12095-015-0142-9

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Official URL: http://dx.doi.org/10.1007/s12095-015-0142-9

Abstract

The Walsh transform Qˆ of a quadratic function Q:Fpn→Fp satisfies |Qˆ|∈{0,pn+s2} for an integer 0 ≤ s ≤ n−1. We study quadratic functions given in trace form Q(x)=Trn(∑ki=0aixpi+1) with the restriction that ai∈Fp, 0≤i≤k. We determine the expected value for the parameter s for such quadratic functions from Fpn to Fp, 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.

Item Type:Article
Uncontrolled Keywords:Quadratic functions; Walsh transform; Expected value; Variance; nonlinearity; Discrete fourier transform
Subjects:Q Science > QA Mathematics > QA150-272.5 Algebra
ID Code:28622
Deposited By:Alev Topuzoğlu
Deposited On:25 Dec 2015 18:04
Last Modified:07 Nov 2016 12:10

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