Discrete Logarithms: The Past and the Future

The first practical public key cryptosystem to be published, the Diffie-Hellman key exchange algorithm, was based on the assumption that discrete logarithms are hard to compute. This intractability hypothesis is also the foundation for the presumed security of a variety of other public key schemes. While there have been substantial advances in discrete log algorithms in the last two decades, in general the discrete log still appears to be hard, especially for some groups, such as those from elliptic curves. Unfortunately no proofs of hardness are available in this area, so it is necessary to rely on experience and intuition in judging what parameters to use for cryptosystems. This paper presents a brief survey of the current state of the art in discrete logs.