A CRT algorithm for constructing genus 2 curves over finite fields

Details A CRT algorithm for constructing genus 2 curves over finite fields A CRT algorithm for constructing genus 2 curves over finite fields

2004-05-15

arxiv.org/abs/math/0405305v2

Mathematics

Number Theory

16 pages. to appear in Proceedings of AGCT-10

Scientific paper

We present a new method for constructing genus 2 curves over a finite field with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discrete-log based cryptosystems. Our algorithm provides an alternative to the traditional CM method for constructing genus 2 curves. For a quartic CM field K with primitive CM type, we compute the Igusa class polynomials modulo p for certain small primes p and then use the Chinese remainder theorem (CRT) and a bound on the denominators to construct the class polynomials. We also provide an algorithm for determining endomorphism rings of ordinary Jacobians of genus 2 curves over finite fields, generalizing the work of Kohel for elliptic curves.

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