On the last fall degree of Weil descent polynomial systems

TL;DR

This paper investigates the relationship between the last fall degrees of polynomial systems and their Weil descent over subfields, providing bounds especially for systems of linearized polynomials.

Contribution

It establishes a theorem linking the last fall degrees of original and Weil descent systems, with applications to linearized polynomial systems.

Findings

Proves a relation between last fall degrees of $\

Provides bounds on last fall degree for Weil descent systems of linearized polynomials.

Abstract

Given a polynomial system $\mathcal{F}$ over a finite field $k$ which is not necessarily of dimension zero, we consider the Weil descent $\mathcal{F}'$ of $\mathcal{F}$ over a subfield $k'$. We prove a theorem which relates the last fall degrees of $\mathcal{F}_1$ and $\mathcal{F}'_1$, where the zero set of $\mathcal{F}_1$ corresponds bijectively to the set of $k$-rational points of $\mathcal{F}$, and the zero set of $\mathcal{F}'_1$ is the set of $k'$-rational points of the Weil descent $\mathcal{F}'$. As an application we derive upper bounds on the last fall degree of $\mathcal{F}'_1$ in the case where $\mathcal{F}$ is a set of linearized polynomials.