**Extended Double-Base Number System with applications to Elliptic Curve Cryptography**

*Christophe Doche and Laurent Imbert*

**Abstract: **We investigate the impact of larger digit sets on the length of
Double-Base Number system (DBNS) expansions.
We present a new representation system called {\em extended DBNS}
whose expansions can be extremely sparse.
When compared with double-base chains, the average length of
extended DBNS expansions of integers of size in the range 200--500 bits
is approximately reduced by $20\%$ using one precomputed point, $30\%$ using two, and $38\%$ using
four.
We also discuss a new approach to approximate an integer $n$ by $d2^a3^b$ where $d$ belongs to a given digit set.
This method, which requires some precomputations as well, leads to realistic DBNS implementations.
Finally, a left-to-right scalar multiplication relying on extended DBNS is given.
On an elliptic curve where operations are performed in Jacobian coordinates, improvements of up to $13\%$
overall can be expected with this approach when compared to window NAF methods
using the same number of precomputed points.
In this context, it is therefore the fastest method known to date to compute a scalar multiplication on a
generic elliptic curve.

**Category / Keywords: **public-key cryptography / Double-Base Number System, Elliptic Curve Cryptography

**Publication Info: **Full version of a paper to appear in the procedding of Indocrypt 2006

**Date: **received 29 Sep 2006

**Contact author: **doche at ics mq edu au

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

**Version: **20061005:144136 (All versions of this report)

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