Paper 2010/156

Genus 2 Curves with Complex Multiplication

Eyal Z. Goren and Kristin E. Lauter


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.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. none
Hyperelliptic Curve CryptographyNumber Theory
Contact author(s)
klauter @ microsoft com
2010-03-24: received
Short URL
Creative Commons Attribution


      author = {Eyal Z.  Goren and Kristin E.  Lauter},
      title = {Genus 2 Curves with Complex Multiplication},
      howpublished = {Cryptology ePrint Archive, Paper 2010/156},
      year = {2010},
      note = {\url{}},
      url = {}
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