Section: New Results

Class invariants in genus 2

Abelian surfaces, or equivalently, Jacobian varieties of genus 2 hyperelliptic curves, offer the same security as elliptic curves in a cryptographic setting and often better efficiency, and could thus be an attractive alternative. The theory of complex multiplication can be used to obtain cryptographically secure curves. Relying on Shimura reciprocity for Siegel modular forms, we have developed the necessary mathematical theory in [24]. It requires deeper algebraic reasoning than for elliptic curves: Ideals of the endomorphism rings of the abelian varieties are no more two-dimensional modules over the integers, but two-dimensional projective modules over quadratic number rings. We succeed in proving results adapted from the elliptic curve case by suitably normalising quadratic forms over number rings and using strong approximation. The result is an elegant theory that leads to clearly formulated and practical algorithms, which we illustrate by examples.