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Explicit Formulas for Real Hyperelliptic Curves of Genus 2 in Affine Representation

  • Stefan Erickson
  • Michael J. JacobsonJr.
  • Ning Shang
  • Shuo Shen
  • Andreas Stein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4547)

Abstract

In this paper, we present for the first time efficient explicit formulas for arithmetic in the degree 0 divisor class group of a real hyperelliptic curve. Hereby, we consider real hyperelliptic curves of genus 2 given in affine coordinates for which the underlying finite field has characteristic > 3. These formulas are much faster than the optimized generic algorithms for real hyperelliptic curves and the cryptographic protocols in the real setting perform almost as well as those in the imaginary case. We provide the idea for the improvements and the correctness together with a comprehensive analysis of the number of field operations. Finally, we perform a direct comparison of cryptographic protocols using explicit formulas for real hyperelliptic curves with the corresponding protocols presented in the imaginary model.

Keywords

hyperelliptic curve reduced divisor infrastructure and distance Cantor’s algorithm explicit formulas efficient implementation cryptographic key exchange 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Stefan Erickson
    • 1
  • Michael J. JacobsonJr.
    • 2
  • Ning Shang
    • 3
  • Shuo Shen
    • 3
  • Andreas Stein
    • 4
  1. 1.Department of Mathematics and Computer Science, Colorado College, 14 E. Cache La Poudre, Colorado Spgs., CO. 80903USA
  2. 2.Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4Canada
  3. 3.Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067USA
  4. 4.Department of Mathematics, University of Wyoming 1000 E. University Avenue, Laramie, WY 82071-3036USA

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