Discrete logarithms: The effectiveness of the index calculus method

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