Galois invariant smoothness basis

Jean-Marc Couveignes (1) , Reynald Lercier (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IRMAR - Institut de Recherche Mathématique de Rennes

This text answers a question raised by Joux and the second author about the computation of discrete logarithms in the multiplicative group of finite fields. Given a finite residue field $\bK$, one looks for a smoothness basis for $\bK^*$ that is left invariant by automorphisms of $\bK$. For a broad class of finite fields, we manage to construct models that allow such a smoothness basis. This work aims at accelerating discrete logarithm computations in such fields. We treat the cases of codimension one (the linear sieve) and codimension two (the function field sieve).

Domaines: Théorie des nombres [math.NT]

Jean-Marc Couveignes, Reynald Lercier. Galois invariant smoothness basis. SAGA, May 2007, Papeete, France. pp.142-167. ⟨hal-00630394⟩