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Genus 2 Curves with Complex Multiplication
Authors: Eyal Z. Goren, Kristin E. LauterAbstract:
Genus 2 curves are useful in cryptography for both discrete-log based and pairing-based systems, but a method is required to compute genus 2 curves with Jacobian with a given number of points. Currently, all known methods involve constructing genus 2 curves with complex multiplication via computing their 3 Igusa class polynomials. These polynomials have rational coefficients and require extensive computation and precision to compute. Both the computation and the complexity analysis of these algorithms can be improved by a more precise understanding of the denominators of the coefficients of the polynomials. The main goal of this paper is to give a bound on the denominators of Igusa class polynomials of genus 2 curves with CM by a primitive quartic CM field K. We give an overview of Igusa’s results on the moduli space of genus two curves and the method to construct genus 2 curves via their Igusa invariants. We also give a complete characterization of the reduction type of a CM abelian surface, for biquadratic, cyclic, and non-Galois quartic CM fields, and for any type of prime decomposition of the prime, including ramified primes.
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