Two Absolutely Irreducible Polynomials over $\Bbb F_2$ and Their Applications to a Conjecture by Carlet
Xiang-dong Hou, Shujun Zhao
TL;DR
This work advances Carlet's conjecture on the sum-freeness of the multiplicative inverse $f_{\text{inv}}$ over $\mathbb{F}_{2^n}$ by establishing the absolute irreducibility of two key polynomials $F_k$ and $\Theta_k$ that encode when a $k$-dimensional $\mathbb{F}_2$-subspace fails to be zero-sum. Building on the known irreducibility of $F_k$ for $k\ge3$, the authors prove $\Theta_k$ is also absolutely irreducible for $k\ge3$ using the Lang-Weil bound and a structural link to $F_k$. This leads to stronger, more general non-sum-free results for $f_{\text{inv}}$, including improved numerical thresholds (e.g., $n\ge 10.8k-15.7$ suffices) and new confirmations for odd $n$ up to $27$, as well as substantial progress when $7\mid n$ with a few possible exceptions. The paper combines algebraic-geometry techniques over finite fields with a ring-theoretic and linear-algebraic view of subspace kernels to enrich the understanding of Carlet's conjecture and to sharpen existing bounds.
Abstract
Two polynomials $F_k(X_1,\dots,X_k)$ and $Θ_k(X_1,\dots,X_k)$ over $\Bbb F_2$ arose from the study of a conjecture by C. Carlet about the sum-freedom of the multiplicative inverse function of $\Bbb F_{2^n}$. Both $F_k$ and $Θ_k$ are homogeneous and symmetric with $\text{deg}\,F_k=2^k-2$ and $\text{deg}\,Θ_k=2^{k-1}$. It is known that $F_k$ is absolutely irreducible for $k\ge 3$. Using the Lang-Weil bound and a curious connection between $F_k$ and $Θ_k$, we show that $Θ_k$ ($k\ge 3$) is also absolutely irreducible. This conclusion allows us to improve several existing results about Carlet's conjecture.