Enumeration of Quadratic Functions With Prescribed Walsh Spectrum

The Walsh transform f̂ of a quadratic function f: F(p
ⁿ
) → 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
^k
a
_i
x
^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.