Group Law Computations on Jacobians of Hyperelliptic Curves

  • Craig Costello
  • Kristin Lauter
Conference paper

DOI: 10.1007/978-3-642-28496-0_6

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7118)
Cite this paper as:
Costello C., Lauter K. (2012) Group Law Computations on Jacobians of Hyperelliptic Curves. In: Miri A., Vaudenay S. (eds) Selected Areas in Cryptography. SAC 2011. Lecture Notes in Computer Science, vol 7118. Springer, Berlin, Heidelberg


We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring \(\mathbb{F}_q[x]\), the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements.

We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form.


Hyperelliptic curves group law Jacobian arithmetic genus 2 

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Craig Costello
    • 1
    • 2
    • 3
  • Kristin Lauter
    • 3
  1. 1.Information Security InstituteQueensland University of TechnologyBrisbaneAustralia
  2. 2.Mathematics DepartmentUniversity of CaliforniaIrvineUSA
  3. 3.Microsoft ResearchRedmondUSA

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