Elliptic Curves and their use in Cryptography

The security of many cryptographic protocols depends on the diiculty of solving the so-called \discrete logarithm" problem, in the multiplica-tive group of a nite eld. Although, in the general case, there are no polynomial time algorithms for this problem, constant improvements are being made { with the result that the use of these protocols require much larger key sizes, for a given level of security, than may be convenient. An abstraction of these protocols shows that they have analogues in any group. The challenge presents itself: nd some other groups for which there are no good attacks on the discrete logarithm, and for which the group operations are suuciently economical. In 1985, the author suggested that the groups arising from a particular mathematical object known as an \elliptic curve" might ll the bill. In this paper I review the general cryptographic protocols which are involved, brieey describe el-liptic curves and review the possible attacks against such cryptosystems. Finally, I mention other applications that elliptic curves have had upon the analysis of other cryptosystems not involving them.