Cryptographically relevant elliptic curves
The contents of this chapter arose from collaboration with Andreas Enge. Following an unpublished idea of Hasse (1933), independently rediscovered by H.M. Stark (1996) and refined by Rubin and Silverberg (2007) and Morain (2007), the reduction of elliptic curves with complex multiplication leads to elliptic curves over finite fields whose cardinality is explicitly predicted. Compared to the choice of elliptic curves at random this method is useful for primality tests and for cryptographic applications based on pairings as, for instance, described by Freeman, Scott and Teske (2006). In particular, this method can be applied to the construction of elliptic curves of prime cardinality.
The approaches ...