Kaşıkcı, Canan and Meidl, Wilfried and Topuzoğlu, Alev (2016) Spectra of a class of quadratic functions: average behaviour and counting functions. Cryptography and Communications (SI), 8 (2). pp. 191214. ISSN 19362447 (Print) 19362455 (Online)
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Official URL: http://dx.doi.org/10.1007/s1209501501429
Abstract
The Walsh transform (Q) over cap of a quadratic function Q : Fp(n) > Fp satisfies vertical bar(Q) over cap vertical bar is an element of {0, p(n+s/2)} for an integer 0 <= s <= n  1. We study quadratic functions given in trace form Q( x) = Trn(Sigma(k)(i=0) a(i)x(pi+1)) with the restriction that a(i) is an element of Fp, 0 <= i <= k. We determine the expected value for the parameter s for such quadratic functions from Fp(n) 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, Topuzo. glu and Meidl, Roy, Topuzo. glu 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 

Additional Information:  Conference: International Workshop on Boolean Functions and their Applications (BFA) / Location: Rosendal, NORWAY / Date: SEP, 2014 
Uncontrolled Keywords:  Quadratic functions; Walsh transform; Expected value; Variance; nonlinearity; Discrete fourier transform 
Subjects:  Q Science > QA Mathematics > QA150272.5 Algebra 
Divisions:  Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences 
Depositing User:  Alev Topuzoğlu 
Date Deposited:  07 Nov 2016 12:15 
Last Modified:  07 Nov 2016 12:15 
URI:  https://research.sabanciuniv.edu/id/eprint/30762 
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Spectra of a class of quadratic functions: Average behaviour and counting functions. (deposited 25 Dec 2015 18:04)
 Spectra of a class of quadratic functions: average behaviour and counting functions. (deposited 07 Nov 2016 12:15) [Currently Displayed]