On quadratic binomial vectorial functions with maximal bent components
2026-04-09 • Information Theory
Information Theory
AI summaryⓘ
The authors study a special type of function over a finite field with size $2^n$, where $n$ is even. They focus on binomial functions with certain properties related to their components called bent components, which are valuable in cryptography. When the function's exponents have limited complexity (their 2-adic weights are at most 2) and some technical conditions hold, the authors show that the function must be equivalent to one of two known simple forms. They also provide bounds on how resistant these functions are to specific cryptographic attacks, using properties of their output values.
finite fieldvectorial Boolean functionbent components2-adic Hamming weightaffine equivalenceFrobenius mapnonlinearitydifferential uniformitybinomial functioncryptographic function
Authors
Xianhong Xie, Yi Ouyang, Shenxing Zhang
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.