Advances in the CM method for elliptic curves

  • Francois Morain | Ecole polytechnique

The complex multiplication method (CM method) builds an algebraic curve over a given finite field GF(q) and having an easily computable cardinality. Used at first for elliptic curves, this method is one of the building blocks of the ECPP algorithm that proves the primality of large integers, and it appeared interesting for other applications, the most recent of which being the construction of pairing friendly curves. The aim of the talk is to recall the method, give some applications, and survey recent advances on several parts of the method, due to various authors, concentrating on elliptic curves. This includes class invariant computations, and the potential use of the Montgomery/Edwards parametrization of elliptic curves.

Speaker Details

F. Morain is Professor at École Polytechnique in France, and head of the Project-Team TANC (Algorithmic Number Theory for Cryptology) of INRIA Saclay Ile de France. His main interests involve the application of elliptic curves to various problems in number theory, including primality proving of large integers (ECPP and fastECPP), and cardinality computations. He holds two records: the largest ordinary number proven prime (> 20,000 decimal digits) and the computation of cardinalities of elliptic curves over large finite fields (2,500 decimal digits), both obtained using distributed computations.

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