Discrete Logarithms : the E ectivenessof the Index Calculus

In this article we survey recent developments concerning the discrete logarithm problem. Both theoretical and practical results are discussed. We emphasize the case of nite elds, and in particular, recent modiications of the index calculus method, including the number eld sieve and the function eld sieve. We also provide a sketch of the some of the cryptographic schemes whose security depends on the intractibility of the discrete logarithm problem.