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arXiv · 2502.04545

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

Xiang-dong Hou·Shujun Zhao
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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.

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Xiang-dong Hou, Shujun Zhao. 2025-02-06. Two Absolutely Irreducible Polynomials over $\Bbb F_2$ and Their Applications to a Conjecture by Carlet. https://arxiv.org/abs/2502.04545

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