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

TL;DR

This paper proves the absolute irreducibility of two polynomials over b2, derived from a conjecture by Carlet, leading to improved results on the conjecture related to sum-freedom of the inverse function.

Contribution

It establishes the absolute irreducibility of b2-polynomials f_k and ftheta_k, connecting their properties to Carlet's conjecture and enhancing previous findings.

Findings

f_k is absolutely irreducible for k

ftheta_k is also absolutely irreducible for k

Results improve understanding of Carlet's conjecture

Abstract

Two polynomials $F_k(X_1,\dots,X_k)$ and $\Theta_k(X_1,\dots,X_k)$ 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 $\Theta_k$ are homogeneous and symmetric with $\text{deg}\,F_k=2^k-2$ and $\text{deg}\,\Theta_k=2^{k-1}$. 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 $\Theta_k$, we show that $\Theta_k$ ($k\ge 3$) is also absolutely irreducible. This conclusion allows us to improve several existing results about Carlet's conjecture.