One or more techniques and/or systems are disclosed for generating a genus 2 curve for use in cryptography. One or more invariant values used to generate the genus 2 curve are determined by evaluating one or more invariant functions on a Hilbert modular surface. The genus 2 curve is generated using the one or more invariant values to determine an equation describing the genus 2 curve. A group is generated from the genus 2 curve, and the group may be used for a cryptographic application.