Enumeration of Quadratic Functions With Prescribed Walsh Spectrum.

The Walsh transform \(\widehat {f}\) of a quadratic function \(f: {\mathbb F}_{p^{n}}\rightarrow {\mathbb F} _{p}\) satisfies \(|\widehat {f}| \in \{0,p^{{n+s}/{2}}\}\) for an integer \(0\le s\le n-1\) , depending on \(f\) . In this paper, quadratic functions of the form \({\mathcal {F}}_{p,n}(x) = {\rm Tr_{n}}(\sum _{i=0}^{k}a_{i}x^{p^{i}+1})\) are studied, with the restriction that \(a_{i} \in {\mathbb F} _{p},~ 0\leq i\leq k\) . 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 \(n=2m\) , for odd \(m\) and \(p=2.\) The number of bent and semibent functions for various classes of \(n\) is also obtained [ABSTRACT FROM AUTHOR]