Minimality and other properties of the width- nonadjacent form

Authors:
James A. Muir and Douglas R. Stinson

Journal:
Math. Comp. **75** (2006), 369-384

MSC (2000):
Primary 94A60, 11T71, 14G50; Secondary 94B40

DOI:
https://doi.org/10.1090/S0025-5718-05-01769-2

Published electronically:
July 12, 2005

MathSciNet review:
2176404

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an integer and let be the set of integers that includes zero and the odd integers with absolute value less than . Every integer can be represented as a finite sum of the form , with , such that of any consecutive 's at most one is nonzero. Such representations are called *width- nonadjacent forms* (-NAFs). When these representations use the digits and coincide with the well-known *nonadjacent forms*. Width- nonadjacent forms are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. We provide some new results on the -NAF. We show that -NAFs have a minimal number of nonzero digits and we also give a new characterization of the -NAF in terms of a (right-to-left) lexicographical ordering. We also generalize a result on -NAFs and show that any base 2 representation of an integer, with digits in , that has a minimal number of nonzero digits is at most one digit longer than its binary representation.

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Additional Information

**James A. Muir**

Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Email:
jamuir@uwaterloo.ca

**Douglas R. Stinson**

Affiliation:
School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Email:
dstinson@uwaterloo.ca

DOI:
https://doi.org/10.1090/S0025-5718-05-01769-2

Keywords:
Efficient representations,
minimal weight,
elliptic curve arithmetic

Received by editor(s):
March 29, 2004

Received by editor(s) in revised form:
August 21, 2004

Published electronically:
July 12, 2005

Additional Notes:
The second author is supported by NSERC grant RGPIN 203114-02.

Article copyright:
© Copyright 2005
American Mathematical Society