A general framework for subexponential discrete logarithm algorithms

Andreas Enge (1) , Pierrick Gaudry (2)

1 Lehrstuhl für Diskrete Mathematik, Optimierung und Operations Research
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

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 finite fields and class groups of number and function fields 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 satisfied. We also show how to modify the algorithm for cyclic subgroups of arbitrary groups when the smoothness assumption can only be verified for the full group.

Domaines: Cryptographie et sécurité [cs.CR]

Andreas Enge, Pierrick Gaudry. A general framework for subexponential discrete logarithm algorithms. Acta Arithmetica, 2002, 102, pp.83-103. ⟨10.4064/aa102-1-6⟩. ⟨inria-00512717⟩