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A CRT algorithm for constructing genus 2 curves over finite fields

Kirsten Eisentraeger·Kristin Lauter
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Abstract

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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Kirsten Eisentraeger, Kristin Lauter. 2007-01-11. A CRT algorithm for constructing genus 2 curves over finite fields. https://arxiv.org/abs/math/0405305

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