Fast genus 2 arithmetic based on Theta functions

Pierrick Gaudry (1, 2)

1 TANC - Algorithmic number theory for cryptology
2 CACAO - Curves, Algebra, Computer Arithmetic, and so On

In 1986, D. V. Chudnovsky and G. V. Chudnovsky proposed to use formulae coming from Theta functions for the arithmetic in Jacobians of genus 2 curves. We follow this idea and derive fast formulae for the scalar multiplication in the Kummer surface associated to a genus 2 curve, using a Montgomery ladder. Our formulae can be used to design very efficient genus 2 cryptosystems that should be faster than elliptic curve cryptosystems in some hardware configurations.

Mots clés: Hyperelliptic curve cryptography; explicit formulae; Kummer surface; theta functions

Domaines: Cryptographie et sécurité [cs.CR]

Pierrick Gaudry. Fast genus 2 arithmetic based on Theta functions. Journal of Mathematical Cryptology, 2007, 1 (3), pp.243-265. ⟨10.1515/JMC.2007.012⟩. ⟨inria-00000625v2⟩