On quadratic binomial vectorial functions with maximal bent components

TL;DR

This paper characterizes quadratic binomial vectorial functions with maximal bent components over finite fields, identifying conditions under which they are affine equivalent to known functions and providing bounds on their nonlinearity and differential uniformity.

Contribution

It proves that such functions are affine equivalent to specific known functions under certain conditions on their exponents and field structure.

Findings

Functions are affine equivalent to known forms under given conditions.

Bounds on nonlinearity and differential uniformity are established.

Conditions involve 2-adic weights and field extension properties.

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 \[ \ell(n):=\min_{\gamma:~\F_2(\gamma)=\F_{2^n}} \dim_{\F_2}\F_2[\sigma]\gamma >m, \] where $\sigma$ 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.