Explicit formulas for real hyperelliptic curves of genus 2 in affine representation

Stefan Erickson; Michael J. Jacobson, Jr.; Andreas Stein

doi:10.3934/amc.2011.5.623

Article Contents

2011, Volume 5, Issue 4: 623-666. Doi: 10.3934/amc.2011.5.623

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Explicit formulas for real hyperelliptic curves of genus 2 in affine representation

1.
Department of Mathematics and Computer Science, Colorado College, 14 E. Cache La Poudre, Colorado Springs, CO 80903
2.
Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4
3.
Institut für Mathematik, Carl-von-Ossietzky Universität Oldenburg, D-26111 Oldenburg

Received: August 31, 2010

Revised: May 31, 2011

Published: October 2011

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Abstract

We present a complete set of efficient explicit formulas for arithmetic in the degree $0$ divisor class group of a genus two real hyperelliptic curve given in affine coordinates. In addition to formulas suitable for curves defined over an arbitrary finite field, we give simplified versions for both the odd and the even characteristic cases. Formulas for baby steps, inverse baby steps, divisor addition, doubling, and special cases such as adding a degenerate divisor are provided, with variations for divisors given in reduced and adapted basis. We describe the improvements and the correctness together with a comprehensive analysis of the number of field operations for each operation. 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.

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Mathematics Subject Classification: Primary: 94A60, 14H45; Secondary: 14Q05.

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