Effective Construction of a Class of Bent Quadratic Boolean Functions

TL;DR

This paper characterizes the bentness of a specific class of quadratic Boolean functions, providing explicit results for cases where the parameter m has particular factorizations, and enumerates functions in a special case.

Contribution

It offers new criteria for determining bentness of quadratic Boolean functions for specific m factorizations and enumerates such functions when m=2^vpq.

Findings

Bentness characterized for m=2^vp^r and m=2^vpq cases.

Explicit enumeration of quadratic bent functions for m=2^vpq.

Provides conditions under which these functions are bent.

Abstract

In this paper, we consider the characterization of the bentness of quadratic Boolean functions of the form $f(x)=\sum_{i=1}^{\frac{m}{2}-1} Tr^n_1(c_ix^{1+2^{ei}})+ Tr_1^{n/2}(c_{m/2}x^{1+2^{n/2}}) ,$ where $n=me$, $m$ is even and $c_i\in GF(2^e)$. For a general $m$, it is difficult to determine the bentness of these functions. We present the bentness of quadratic Boolean function for two cases: $m=2^vp^r$ and $m=2^vpq$, where $p$ and $q$ are two distinct primes. Further, we give the enumeration of quadratic bent functions for the case $m=2^vpq$.