Enumeration of Quadratic Functions With Prescribed Walsh Spectrum

The Walsh transform f̂ of a quadratic function f: F(pⁿ) → F_p satisfies |f̂| ∈ {0,p^n+s/2} for an integer 0 ≤ s ≤ n-1, depending on f. In this paper, quadratic functions of the form F_p,n(x) = Tr_n(Σ_i=0^ka_ix^pi+1) are studied, with the restriction that a_i ∈ F_p, 0 ≤ i ≤ k. Three methods for enumeration of such functions are presented when the value for s is prescribed. This paper extends earlier enumeration results significantly, for instance, the generating function for the counting function is obtained, when n is odd and relatively prime to p, or when n = 2 m, for odd m and p = 2. The number of bent and semibent functions for various classes of n is also obtained.