Two Absolutely Irreducible Polynomials over $\Bbb F_2$ and Their Applications to a Conjecture by Carlet

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.