E LI X LABORATOIRE D ' INFORMATIQUE Unité mixte CNRS n ° 7650

We describe a generic algorithm for computing discrete logarithms in groups of known order in which a smoothness concept is available. The running time of the algorithm can be proved without using any heuristics and leads to a subexponential complexity in particular for nite elds and class groups of number and function elds which were proposed for use in cryptography. In class groups, our algorithm is substantially faster than previously suggested ones. The subexponential complexity is obtained for cyclic groups in which a certain smoothness assumption is satissed. We also show how to modify the algorithm for cyclic subgroups of arbitrary groups when the smoothness assumption can only be veriied for the full group. Mots cl es: logarithme discret, calcul d'index, groupes de classes, sous-exponentialit e.