Computing isomorphisms and embeddings of finite fields

Abstract

Let F q be a finite field. Given two irreducible polynomials f, g over F q , with deg f dividing deg g, the finite field embedding problem asks to compute an explicit description of a field embedding of. When deg f = deg g, this is also known as the isomorphism problem.This problem, a special instance of polynomial factorization, plays a central role in computer algebra software. We review previous algorithms, due to Lenstra, Allombert, Rains, and Narayanan, and propose improvements and generalizations. Our detailed complexity analysis shows that our newly proposed variants are at least as efficient as previously known algorithms, and in many cases significantly better.We also implement most of the presented algorithms, compare them with the state of the art computer algebra software, and make the code available as open source. Our experiments show that our new variants consistently outperform available software.

Standard Lattices of Compatibly Embedded Finite Fields

“…x ∈ F p l it is possible to compute its image ϕ(x) using O m (ω+1)/2) operations; similarly, given an element = ϕ(x) in F p m , it is possible to recover x in the same asymptotic number of operations. The relevant algorithms are summarized in [7,Sec. 6]; note that, for specially constructed generators α l , more efficient algorithms may exist [11][12][13][14].…”