**Hybrid Binary-Ternary Joint Sparse Form and its Application in Elliptic Curve Cryptography**

*Jithra Adikari and Vassil Dimitrov and Laurent Imbert*

**Abstract: **Multi-exponentiation is a common and time consuming operation in public-key cryptography. Its elliptic curve counterpart, called multi-scalar multiplication is extensively used for digital signature verification. Several algorithms have been proposed to speed-up those critical computations. They are based on simultaneously recoding a set of integers in order to minimize the number of general multiplications or point additions. When signed-digit recoding techniques can be used, as in the world of elliptic curves, Joint Sparse Form (JSF) and interleaving $w$-NAF are the most efficient algorithms. In this paper, a novel recoding algorithm for a pair of integers is proposed, based on a decomposition that mixes powers of 2 and powers of 3. The so-called Hybrid Binary-Ternary Joint Sparse Form require fewer digits and is sparser than the JSF and the interleaving $w$-NAF. Its advantages are illustrated for elliptic curve double-scalar multiplication; the operation counts show a gain of up to 18\%.

**Category / Keywords: **Multi-exponentiation, Multi-scalar multiplication, Joint sparse form, Binary-ternary number system, Elliptic curves.

**Date: **received 25 Jun 2008, last revised 3 Jul 2008

**Contact author: **jithra adikari at atips ca

**Available format(s): **PDF | BibTeX Citation

**Note: **Major correction in the theoretical analysis

**Version: **20080703:200028 (All versions of this report)

**Short URL: **ia.cr/2008/285

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