Section: Software and Platforms

Finite Fields

Participants : Pierrick Gaudry, Emmanuel Thomé [contact] , Luc Sanselme.

mp${𝔽}_q$ is (yet another) library for computing in finite fields. The purpose of mp ${𝔽}_q$ is not to provide a software layer for accessing finite fields determined at runtime within a computer algebra system like Magma, but rather to give a very efficient, optimized code for computing in finite fields precisely known at compile time. mp ${𝔽}_q$ can adapt to finite fields of any characteristic and any extension degree. However, one of the targets being the use in cryptology, mp ${𝔽}_q$ somehow focuses on prime fields and on fields of characteristic two.

When it was first written in 2007, mp ${𝔽}_q$ established reference marks for fast elliptic curve cryptography: the authors improved over the fastest examples of key-sharing software in genus 1 and 2, both over binary fields and prime fields. A stream of academic works followed the idea behind mp ${𝔽}_q$ and improved over such timings, notably by Scott, Aranha, Longa, Bos, Hisil, Costello.

The library's purpose being the generation of code rather than its execution, the working core of mp ${𝔽}_q$ consists of roughly 18,000 lines of Perl code, which generate most of the C code. mp ${𝔽}_q$ is distributed at http://mpfq.gforge.inria.fr/ .

In 2013, version 1.1 of mp ${𝔽}_q$ has been released. This new release includes new assembly code by Luc Sanselme providing optimized arithmetic over fields whose characteristic fits in a number of bits which fit within half-word boundaries.

In 2013, Hamza Jeljeli collaborated with Bastien Vialla from LIRMM, Montpellier to integrate experimental code based on RNS arithmetic (Residue Number System), intending to provide back-end functionality for the linear algebra code in Cado-nfs . This feature set is still experimental.