Advances in Mathematics of Communications (AMC)

Empirical optimization of divisor arithmetic on hyperelliptic curves over $\mathbb{F}_{2^m}$

Pages: 485 - 502, Volume 7, Issue 4, November 2013      doi:10.3934/amc.2013.7.485

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Laurent Imbert - CNRS, LIRMM, Université Montpellier 2, 161 rue Ada, F-34095 Montpellier, France (email)
Michael J. Jacobson, Jr. - Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada (email)

Abstract: A significant amount of effort has been devoted to improving divisor arithmetic on low-genus hyperelliptic curves via explicit versions of generic algorithms. Moderate and high genus curves also arise in cryptographic applications, for example, via the Weil descent attack on the elliptic curve discrete logarithm problem, but for these curves, the generic algorithms are to date the most efficient available. Nagao [22] described how some of the techniques used in deriving efficient explicit formulas can be used to speed up divisor arithmetic using Cantor's algorithm on curves of arbitrary genus. In this paper, we describe how Nagao's methods, together with a sub-quadratic complexity partial extended Euclidean algorithm using the half-gcd algorithm can be applied to improve arithmetic in the degree zero divisor class group. We present numerical results showing which combination of techniques is more efficient for hyperelliptic curves over $\mathbb{F}_{2^n}$ of various genera.

Keywords:  Hyperelliptic curve, divisor addition, Cantor’s algorithm, NUCOMP, half-gcd algorithm.
Mathematics Subject Classification:  Primary: 94A60, 14H45; Secondary: 14Q05.

Received: December 2012;      Available Online: October 2013.