On the Characteristic Polynomial of Linearized Polynomials

TL;DR

This paper introduces an efficient algorithm for computing the characteristic polynomial of linearized polynomials over finite fields, significantly improving computational complexity for large extensions.

Contribution

It presents a novel algorithm with reduced complexity for calculating characteristic polynomials of linearized polynomials over finite fields.

Findings

Algorithm has complexity $O(n(\log(n))^4)$ in $\\mathbb{F}_q$ operations.

Provides a square root speedup over generic algorithms for low-degree polynomial representations.

Applicable to large extension fields with practical efficiency improvements.

Abstract

Let $k$ be a finite field, and $L$ be a $q$-linearized polynomial defined over $k$ of $q$-degree $r$ ($L=\sum^r_{i=0}a_iZ^{q^i}$, with $a_i\in k$). This paper provides an algorithm to compute a characteristic polynomial of $L$ over a large extension field $\mathbb F_{q^n}\supseteq k$. Our algorithm has computational complexity of $O(n(\log(n))^4)$ in terms of $\mathbb F_q$ operations with the implied constant depending only on $k$ and $r$. Up to logarithmic factors, and for linear maps represented by low degree polynomials, this provides a square root improvement over generic algorithms.