Constructing hyperelliptic curves of genus 2 suitable for cryptography

Abstract

Abstract. In this article we show how to generalize the CM-method for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation.

Generating pairing-friendly parameters for the CM construction of genus 2 curves over prime fields

“…⊓ ⊔ Depending on p and O K , there are 2 or 4 possibilities for the group order #Jac(C, F q ) [24] [5]. However, for a demonstration purpose, in the algorithm above we are only interested in curves C whose Jacobian has exact group order given by…”

“…An alternative to point counting is to use the genus 2 Complex Multiplication (CM) algorithm ( [24]) to construct curves with a given number of points on its Jacobian. Like the case of the elliptic curve CM method, the genus 2 CM method is very efficient once the class polynomials of the CM field are computed.…”

“…Find a prime p such that there exists ω ∈ K with ωω = p, and an integer N depending on p and O K which will be the group order of the Jacobian of the genus 2 curve having CM by O K . Such p and N can be identified by using a method in [24]. 2.…”

“…2. Compute the Igusa class polynomials H i (x), i = 1, 2, 3 of K. This step can be done using the methods as described in one of [22], [24], [5], [13]. 3.…”

Abelian Varieties with Prescribed Embedding Degree

“…The class polynomials for K can be found in the preprint version of [13]. We used the roots of the class polynomials mod q to construct curves over F q with CM by O K .…”