Examples of genus two CM curves defined over the rationals
Mathematics of Computation
Analysis of PSLQ, an integer relation finding algorithm
Mathematics of Computation
Construction of Hyperelliptic Curves with CM and Its Application to Cryptosystems
ASIACRYPT '00 Proceedings of the 6th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Action of Modular Correspondences around CM Points
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
Efficient doubling on genus two curves over binary fields
SAC'04 Proceedings of the 11th international conference on Selected Areas in Cryptography
EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
Constructing pairing-friendly genus 2 curves with ordinary Jacobians
Pairing'07 Proceedings of the First international conference on Pairing-Based Cryptography
Generating pairing-friendly parameters for the CM construction of genus 2 curves over prime fields
Designs, Codes and Cryptography
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The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method as far as possible. We have thus designed a new algorithm for the construction of CM invariants of genus 2 curves, using 2-adic lifting of an input curve over a small finite field. This provides a numerically stable alternative to the complex analytic method in the first phase of the CM method for genus 2. As an example we compute an irreducible factor of the Igusa class polynomial system for the quartic CM field ℚ (i√(75 + 12√(17))), whose class number is 50. We also introduce a new representation to describe the CM curves: a set of polynomials in (j1,j2,j3) which vanish on the precise set of triples which are the Igusa invariants of curves whose Jacobians have CM by a prescribed field. The new representation provides a speedup in the second phase, which uses Mestre's algorithm to construct a genus 2 Jacobian of prime order over a large prime field for use in cryptography.