Section: Scientific Foundations
Complex multiplication
Despite the achievements described above, random curves are sometimes difficult to use, since their cardinality is not easy to compute or useful instances are too rare to occur (curves for pairings for instance). In some cases, curves with special properties can be used. For instance curves with complex multiplication (in brief CM), whose cardinalities are easy to compute. For example, the elliptic curve defined over GF(p) of equation y2 = x3 + x has cardinality p + 1-2u , when p = u2 + v2 , and computing u is easy.
The CM theory for genus 1 is well known and dates back to the middle of the nineteenth century (Kronecker, Weber, etc.). Its algorithmic part is also well understood, and recently more work was done, largely by TANC . Twenty years ago, this theory was applied by Atkin to the primality proving of arbitrary integers, yielding the ECPP algorithm developed ever since by F. Morain. Though the decision problem isPrime? was shown to be in P (by the 2002 work of Agrawal, Kayal, Saxena), practical primality proving is still done only with ECPP.
These CM curves enabled A. Enge, R. Dupont and F. Morain to give an algorithm for building good curves that can be used in identity based cryptosystems (cf. infra).
CM curves are defined by algebraic integers, whose minimal polynomial has to be computed exactly, its coefficients being exact integers. The fastest algorithm to perform these computations requires a floating point evaluation of the roots of the polynomial to a high precision. F. Morain on the one hand and A. Enge (together with R. Schertz) on the other, have developed the use of new class invariants that characterize CM curves. The union of these two families is currently the best that can be achieved in the field (see [32] ). Later, F. Morain and A. Enge have designed a fast method for the computation of the roots of this polynomial over a finite field using Galois theory [33] . These invariants, together with this new algorithm, are incorporated in the working version of the program ECPP.
The theory of Complex Multiplication also exists for non-elliptic curves, but is more intricate, and only recently can we dream to use them. The first results in that direction are described below.
