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

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

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.