4 research outputs found
Fast Arithmetics in Artin-Schreier Towers over Finite Fields
An Artin-Schreier tower over the finite field F_p is a tower of field
extensions generated by polynomials of the form X^p - X - a. Following Cantor
and Couveignes, we give algorithms with quasi-linear time complexity for
arithmetic operations in such towers. As an application, we present an
implementation of Couveignes' algorithm for computing isogenies between
elliptic curves using the p-torsion.Comment: 28 pages, 4 figures, 3 tables, uses mathdots.sty, yjsco.sty Submitted
to J. Symb. Compu
Computing in Algebraic Closures of Finite Fields
We present algorithms to construct and perform computations in algebraic closures of finite fields. Inspired by algorithms for constructing irreducible polynomials, our approach for constructing closures consists of two phases; First, extension towers of prime power degree are built, and then they are glued together using composita techniques. To be able to move elements around in the closure we give efficient algorithms for computing isomorphisms and embeddings. In most cases, our algorithms which are based on polynomial arithmetic, rather than linear algebra, have quasi-linear complexity
Change of basis for m-primary ideals in one and two variables
Following recent work by van der Hoeven and Lecerf (ISSAC 2017), we discuss
the complexity of linear mappings, called untangling and tangling by those
authors, that arise in the context of computations with univariate polynomials.
We give a slightly faster tangling algorithm and discuss new applications of
these techniques. We show how to extend these ideas to bivariate settings, and
use them to give bounds on the arithmetic complexity of certain algebras.Comment: In Proceedings ISSAC'19, ACM, New York, USA. See proceedings version
for final formattin