Minimality and other properties of the width nonadjacent form
Authors:
James A. Muir and Douglas R. Stinson
Journal:
Math. Comp. 75 (2006), 369384
MSC (2000):
Primary 94A60, 11T71, 14G50; Secondary 94B40
Published electronically:
July 12, 2005
MathSciNet review:
2176404
Fulltext PDF Free Access
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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 wellknown 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 (righttoleft) 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:
http://dx.doi.org/10.1090/S0025571805017692
PII:
S 00255718(05)017692
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 20311402.
Article copyright:
© Copyright 2005
American Mathematical Society
