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Group law computations on Jacobians of hyperelliptic Curves

Costello, Craig & Lauter, Kristin (2012) Group law computations on Jacobians of hyperelliptic Curves. Lecture Notes in Computer Science, 7118, pp. 92-117.

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    Abstract

    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 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.

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    ID Code: 48857
    Item Type: Journal Article
    Additional Information: Paper presented in Selected Areas in Cryptography (SAC2011), Ryerson University, Ontario, Canada, August 11-12, 2011.
    Keywords: hyperelliptic curves, group law, Jacobian arithmetic, genus 2
    DOI: 10.1007/978-3-642-28496-0_6
    ISSN: 0302-9743
    Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > PURE MATHEMATICS (010100) > Algebra and Number Theory (010101)
    Australian and New Zealand Standard Research Classification > INFORMATION AND COMPUTING SCIENCES (080000) > COMPUTATION THEORY AND MATHEMATICS (080200) > Analysis of Algorithms and Complexity (080201)
    Australian and New Zealand Standard Research Classification > INFORMATION AND COMPUTING SCIENCES (080000) > OTHER INFORMATION AND COMPUTING SCIENCES (089900) > Information and Computing Sciences not elsewhere classified (089999)
    Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
    Past > Institutes > Information Security Institute
    Copyright Owner: Copyright Springer-Verlag Berlin Heidelberg 2012
    Copyright Statement: The original publication is available at SpringerLink http://www.springerlink.com
    Deposited On: 27 Feb 2012 08:15
    Last Modified: 28 Feb 2012 05:20

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