A class of functions with low-valued Walsh spectrum

TL;DR

This paper studies the Walsh spectrum of specific monomial functions over finite fields, providing explicit value distributions in certain cases related to prime powers and class numbers, which aids in understanding their cryptographic properties.

Contribution

It explicitly determines the Walsh spectrum distribution of monomial functions with index two under particular prime and class number conditions.

Findings

Explicit Walsh spectrum distribution for specific monomial functions.

Connection between Walsh spectrum values and class numbers of quadratic fields.

Enhanced understanding of functions with low Walsh spectrum for cryptographic applications.

Abstract

Let $l\equiv 3\pmod 4$, $l\ne 3$, be a prime, $N=l^2$, $f=\frac{l(l-1)}2$ the multiplicative order of a prime $p$ modulo $N$, and $q=p^f$. In this paper, we investigate the Walsh spectrum of the monomial functions $f(x)={\rm Tr}_{q/p}(x^{\frac{q-1}{l^2}})$ in index two case. In special, we explicitly present the value distribution of the Walsh transform of $f(x)$ if $1+l=4p^h$, where $h$ is a class number of $\Bbb Q(\sqrt{-l})$.