The Past, Evolving Present, and Future of the Discrete Logarithm

Abstract

The first practical public key cryptosystem ever published, the Diffie-Hellman key exchange algorithm, relies for its security on the assumption that discrete logarithms are hard to compute. This intractability hypothesis is also the foundation for the security of a large variety of other public key systems and protocols. Since the introduction of the Diffie-Hellman key exchange more than three decades ago, there have been substantial algorithmic advances in the computation of discrete logarithms. However, in general the discrete logarithm problem is still considered to be hard. In particular, this is the case for the multiplicative group of finite fields with medium to large characteristic and for the additive group of a general elliptic curve. This paper presents a current survey of the state of the art concerning discrete logarithms and their computation. 1