Standard Lattices of Compatibly Embedded Finite Fields

Luca de Feo (1) , Hugues Randriambololona (2) , Édouard Rousseau (2, 1)

1 LMV - Laboratoire de Mathématiques de Versailles
2 LTCI - Laboratoire Traitement et Communication de l'Information

Lattices of compatibly embedded finite fields are useful in computer algebra systems for managing many extensions of a finite field F_p at once. They can also be used to represent the algebraic closure F̄_p , and to represent all finite fields in a standard manner. The most well known constructions are Conway polynomials, and the Bosma–Cannon–Steel framework used in Magma. In this work, leveraging the theory of the Lenstra-Allombert isomorphism algorithm, we generalize both at the same time. Compared to Conway polynomials, our construction defines a much larger set of field extensions from a small pre-computed table; however it is provably as inefficient as Conway polynomials if one wants to represent all field extensions, and thus yields no asymptotic improvement for representing F̄_p . Compared to Bosma–Cannon–Steel lattices, it is considerably more efficient both in computation time and storage: all algorithms have at worst quadratic complexity, and storage is linear in the number of represented field extensions and their degrees. Our implementation written in C/Flint/Julia/Nemo shows that our construction in indeed practical.

Domains: Number Theory [math.NT]; Symbolic Computation [cs.SC]

Luca de Feo, Hugues Randriambololona, Édouard Rousseau. Standard Lattices of Compatibly Embedded Finite Fields. ISSAC 2019, Jul 2019, Beijing, China. ⟨10.1145/3326229.3326251⟩. ⟨hal-02136976⟩