Journal of Cryptology
Reducing elliptic curve logarithms to logarithms in a finite field
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves
Mathematics of Computation
A course in computational algebraic number theory
A course in computational algebraic number theory
Examples of genus two CM curves defined over the rationals
Mathematics of Computation
Efficient construction of secure hyperelliptic discrete logarithm problems
ICICS '97 Proceedings of the First International Conference on Information and Communication Security
Speeding up the Discrete Log Computation on Curves with Automorphisms
ASIACRYPT '99 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
Fast Construction of Secure Discrete Logarithm Problems over Jacobian Varieties
Proceedings of the IFIP TC11 Fifteenth Annual Working Conference on Information Security for Global Information Infrastructures
Computing in the jacobian of a plane algebraic curve
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
Counting Rational Points on Curves and Abelian Varieties over Finite Fields
ANTS-II Proceedings of the Second International Symposium on Algorithmic Number Theory
A p-adic algorithm for univariate partial fractions
SYMSAC '81 Proceedings of the fourth ACM symposium on Symbolic and algebraic computation
Counting rational points on curves over finite fields
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
The 2-adic CM method for genus 2 curves with application to cryptography
ASIACRYPT'06 Proceedings of the 12th international conference on Theory and Application of Cryptology and Information Security
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Construction of secure hyperelliptic curves is of most important yet most difficult problem in design of cryptosystems based on the discrete logarithm problems on hyperelliptic curves. Presently the only accessible approach is to use CM curves. However, to find models of the CM curves is nontrivial. The popular approach uses theta functions to derive a projective embedding of the Jacobian varieties, which needs to calculate the theta functions to very high precision. As we show in this paper, it costs computation time of an exponential function in the discriminant of the CM field. This paper presents new algorithms to find explicit models of hyperelliptic curves with CM. Algorithms for CM test of Jacobian varieties of algebraic curves and to lift from small finite fields both the models and the invariants of CM curves are presented. We also show that the proposed algorithm for invariants lifting has complexity of a polynomial time in the discriminant of the CM field.