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

S.  Erickson; M.  J.  Jacobson Jr.; A.  Stein

Paper 2011/105

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

S. Erickson, M. J. Jacobson Jr., and A. Stein

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.

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Published elsewhere. Submitted to Advances in Mathematics of Communication
Keywords: hyperelliptic curve reduced divisor infrastructure and distance Cantor’s algorithm explicit formulas efficient implementation cryptographic key exchange
Contact author(s): jacobs @ cpsc ucalgary ca
History: 2011-03-05: received
Short URL: https://ia.cr/2011/105
License: CC BY

BibTeX

@misc{cryptoeprint:2011/105,
      author = {S.  Erickson and M.  J.  Jacobson Jr. and A.  Stein},
      title = {Explicit Formulas for Real Hyperelliptic Curves of Genus 2 in Affine Representation},
      howpublished = {Cryptology {ePrint} Archive, Paper 2011/105},
      year = {2011},
      url = {https://eprint.iacr.org/2011/105}
}