Basically, the ElGamal public key algorithm using elliptic curve groups over finite fields. This is important because it seems that there are no known subexponential algorithms for computing discrete logarithms in elliptic curve groups, as exist in other groups. No definition of smoothness exists in elliptic curve groups so the more efficient methods for computing discrete logarithms that have been developed (e.g. the number field sieve) cannot be used, or so it seems given the present (2001) state of mathematical knowledge. This means that it's probably possible to get away with using much smaller key sizes with elliptic curve cryptography than with other methods.

Common finite fields used for elliptic curve cryptography are GF(p) or GF(2m) with a polynomial representation or an optimal normal basis representation.

Certicom is the company that has been promoting much of the research and use of this algorithm, and they hold patents on the efficient methods for implementing this. Like RSA's factoring contest they also have a contest for computing discrete logarithms in elliptic curve groups.

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