## Cryptology ePrint Archive: Report 2005/228

**Efficient Doubling on Genus 3 Curves over Binary Fields**

*Xinxin Fan and Thomas Wollinger and Yumin Wang*

**Abstract: **The most important and expensive operation in a hyperelliptic curve
cryptosystem (HECC) is scalar multiplication by an integer k, i.e., computing an integer k times a divisor D on the Jacobian. Using some recoding algorithms for scalar $k$, we can reduce a number of divisor class additions during the process of computing scalar multiplication. So divisor doubling will account for the main part in all kinds of scalar multiplication algorithms. In order to accelerate the genus 3 HECC over binary fields we investigate how to
compute faster doubling in this paper.

By constructing birational transformation of variables, we derive
explicit doubling formulae for all types of defining equations of
the curve. For each type of curve, we analyze how many field operations are needed. So far all proposed curves are secure,
though they are more special types. Our results allow to choose
curves from a large enough variety which have extremely fast
doubling needing only one third the time of an addition in the
best case. Furthermore, an actual implementation of the new formulae
on a Pentium-M processor shows its practical relevance.

**Category / Keywords: **implementation / Hyperelliptic Curve Cryptosystem

**Date: **received 14 Jul 2005

**Contact author: **xxfan at mail xidian edu cn

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Version: **20050720:062727 (All versions of this report)

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