On quadratic binomial vectorial functions with maximal bent components

2026-04-09

AI Summary

This work investigates quadratic binomial vectorial Boolean functions over finite fields of the form \( F(x) = x^{d_1} + x^{d_2} \), under the conditions that they possess the maximum possible number of bent components and that the 2-adic Hamming weights of the exponents \( d_1 \) and \( d_2 \) are at most two. By integrating tools from finite field theory, Frobenius automorphisms, and affine equivalence analysis, the study establishes that such functions are affinely equivalent to only two canonical forms, thereby proving structural uniqueness. Furthermore, it derives the first tight upper and lower bounds on nonlinearity and differential uniformity based on the size of the image set, offering novel theoretical foundations for the design of cryptographic functions.

Abstract

Assume $n=2m\geq 2$ and let $F(x)=x^{d_1}+x^{d_2}$ be a binomial vectorial function over $\F_{2^n}$ possessing the maximal number (i.e. $2^n-2^m$) of bent components. Suppose the $2$-adic Hamming weights $\wt_2(d_1)$ and $\wt_2(d_2)$ are both at most $2$, we prove that $F(x)$ is affine equivalent to either $x^{2^m+1}$ or $x^{2^i}(x+x^{2^m})$, provided that \[ \ell(n):=\min_{γ:~\F_2(γ)=\F_{2^n}} \dim_{\F_2}\F_2[σ]γ>m, \] where $σ$ is the Frobenius $(x\mapsto x^2)$ on $\F_{2^n}$, and $\gcd(d_1,d_2,2^m-1)>1$. Under this condition, we also establish two bounds on the nonlinearity and the differential uniformity of $F$ by means of the cardinality of its image set.

Problem

• bent components
• quadratic binomial functions
• vectorial Boolean functions
• affine equivalence
• nonlinearity

Innovation

• bent components
• quadratic binomial functions
• affine equivalence
• nonlinearity
• differential uniformity