Genus 2 point counting over prime fields

• Published in: Journal of symbolic computation Vol. 47; no. 4; pp. 368 - 400
• Main Authors: Gaudry, Pierrick, Schost, Éric
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01-04-2012; Elsevier
• Subjects: Computer Science; Cryptography and Security; Hyperelliptic curves; Point counting; Schoof–Pila algorithm; Hyperelliptic curves; Point counting; Schoof–Pila algorithm
• ISSN: 0747-7171; 1095-855X
• DOI: 10.1016/j.jsc.2011.09.003

For counting points of Jacobians of genus 2 curves over a large prime field, the best known approach is essentially an extension of Schoof’s genus 1 algorithm. We propose various practical improvements to this method and illustrate them with a large scale computation: we counted hundreds of curves, until one was found that is suitable for cryptographic use, with a state-of-the-art security level of approximately 2128 and desirable speed properties. This curve and its quadratic twist have a Jacobian group whose order is 16 times a prime.
► We give a detailed point-counting algorithm for genus 2 curves over a prime field. ► We conduct a large-scale computation, using more than 1000,000 CPU hours. ► We obtain a doubly-secure curve with 128 bit security. ► This curve has small coefficients for the Kummer pseudo-group law.