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Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension

Anbar, Nurdagül and Meidl, Wilfried and Topuzoğlu, Alev Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension. Designs, Codes and Cryptography . ISSN 0925-1022 (Print) 1573-7586 (Online) Published Online First http://dx.doi.org/10.1007/s10623-016-0213-8

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Official URL: http://dx.doi.org/10.1007/s10623-016-0213-8

Abstract

The Walsh transform QˆQ^ of a quadratic function Q:Fpn→FpQ:Fpn→Fp satisfies |Qˆ(b)|∈{0,pn+s2}|Q^(b)|∈{0,pn+s2} for all b∈Fpnb∈Fpn , where 0≤s≤n−10≤s≤n−1 is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class C1C1 is defined for arbitrary n as C1={Q(x)=Trn(∑⌊(n−1)/2⌋i=1aix2i+1):ai∈F2}C1={Q(x)=Trn(∑i=1⌊(n−1)/2⌋aix2i+1):ai∈F2} , and the larger class C2C2 is defined for even n as C2={Q(x)=Trn(∑(n/2)−1i=1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2}C2={Q(x)=Trn(∑i=1(n/2)−1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2} . For an odd prime p, the subclass DD of all p-ary quadratic functions is defined as D={Q(x)=Trn(∑⌊n/2⌋i=0aixpi+1):ai∈Fp}D={Q(x)=Trn(∑i=0⌊n/2⌋aixpi+1):ai∈Fp} . We determine the generating function for the distribution of the parameter s for C1,C2C1,C2 and DD . As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case p>2p>2 , the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to C1C1 and C2C2 in terms of a generating function.

Item Type:Article
Uncontrolled Keywords:Quadratic functions, Plateaued functions, Bent functions, Walsh transform, Idempotent functions, Rotation symmetric, Reed-Muller code
Subjects:Q Science > QA Mathematics > QA150-272.5 Algebra
ID Code:30811
Deposited By:Alev Topuzoğlu
Deposited On:07 Nov 2016 12:05
Last Modified:07 Nov 2016 12:05

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