Fast Computation of Isomorphisms Between Finite Fields Using Elliptic Curves

TL;DR

This paper introduces a randomized algorithm for computing finite field isomorphisms using elliptic curves, achieving subquadratic time complexity for many cases, improving over previous quadratic-time methods.

Contribution

The paper presents a novel randomized algorithm leveraging elliptic curves to compute finite field isomorphisms more efficiently, especially when the degree has no large prime factors.

Findings

Achieves subquadratic time complexity for many degrees n.

Provides a nearly linear time complexity for degrees n with no large prime factors.

Formulates an open problem related to finding points on elliptic curves of prescribed order.

Abstract

We propose a randomized algorithm to compute isomorphisms between finite fields using elliptic curves. To compute an isomorphism between two fields of cardinality $q^n$, our algorithm takes $$n^{1+o(1)} \log^{1+o(1)}q + \max_{\ell} \left(\ell^{n_\ell + 1+o(1)} \log^{2+o(1)} q + O(\ell \log^5q)\right)$$ time, where $\ell$ runs through primes dividing $n$ but not $q(q-1)$ and $n_\ell$ denotes the highest power of $\ell$ dividing $n$. Prior to this work, the best known run time dependence on $n$ was quadratic. Our run time dependence on $n$ is at worst quadratic but is subquadratic if $n$ has no large prime factor. In particular, the $n$ for which our run time is nearly linear in $n$ have natural density at least $3/10$. The crux of our approach is finding a point on an elliptic curve of a prescribed prime power order or equivalently finding preimages under the Lang map on elliptic curves over finite fields. We formulate this as an open problem whose resolution would solve the finite field isomorphism problem with run time nearly linear in $n$.