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

Laurent Imbert 1, * Michael Jacobson Jr 2
* Corresponding author
1 ECO - Exact Computing
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
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~\cite{Nagao2000} 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.
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Contributor : Laurent Imbert <>
Submitted on : Monday, October 7, 2013 - 11:00:12 AM
Last modification on : Tuesday, December 11, 2018 - 5:16:02 PM

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Laurent Imbert, Michael Jacobson Jr. Empirical optimization of divisor arithmetic on hyperelliptic curves over $\mathbf{F}_{2^m}$. Advances in Mathematics of Communications, AIMS, 2013, 7 (4), pp.485-502. ⟨10.3934/amc.2013.7.485⟩. ⟨lirmm-00870376⟩

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