Taking Roots over High Extensions of Finite Fields

TL;DR

This paper introduces a new subquadratic algorithm for computing m-th roots over finite fields, significantly improving efficiency especially for the case m=2, with practical implications for finite field computations.

Contribution

The paper presents a novel algorithm for m-th root extraction in finite fields with improved complexity, particularly for square roots, using polynomial multiplication and composition techniques.

Findings

Expected complexity for square root case is O(M(n) log p + C(n) log n) operations in _p

The algorithm is subquadratic in n, with specific bounds for polynomial multiplication and composition

Practical efficiency gains over previous methods in finite field root extraction

Abstract

We present a new algorithm for computing $m$-th roots over the finite field $\F_q$, where $q = p^n$, with $p$ a prime, and $m$ any positive integer. In the particular case $m=2$, the cost of the new algorithm is an expected $O(\M(n)\log (p) + \CC(n)\log(n))$ operations in $\F_p$, where $\M(n)$ and $\CC(n)$ are bounds for the cost of polynomial multiplication and modular polynomial composition. Known results give $\M(n) = O(n\log (n) \log\log (n))$ and $\CC(n) = O(n^{1.67})$, so our algorithm is subquadratic in $n$.