**A Simple Left-to-Right Algorithm for the Computation of the Arithmetic Weight of Integers**

*James A. Muir*

**Abstract: **We present a simple algorithm for computing the arithmetic weight of an integer with respect to a given radix r>=2. The arithmetic weight of n is the minimum number of nonzero digits in any signed radix-r representation of n. This algorithm leads to a new family of minimal weight signed radix-r representations which can be constructed using a
left-to-right on-line algorithm. These representations are different from the ones previously reported by Joye and Yen at PKC 2002. The idea behind our algorithm is that of choosing closest elements which was introduced by Muir and Stinson at CT-RSA 2005. Our results have applications in coding theory and in the efficient implementation of public-key cryptography.

**Category / Keywords: **implementation / signed radix-r representations, left-to-right recoding, elliptic curve arithmetic

**Date: **received 7 Jan 2006, last revised 23 Oct 2006

**Contact author: **jamuir at scs carleton ca

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

**Note: **This new version corrects an error that I overlooked related to the difference between making a closest choice for even radices and odd radices. All the algorithms are the same; it's just that now the proof of minimality is correct :-)

**Version: **20061023:220526 (All versions of this report)

**Short URL: **ia.cr/2006/008

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