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Journal of Mathematical Cryptology

Managing Editor: Magliveras, Spyros S. / Steinwandt, Rainer / Trung, Tran

Editorial Board: Blackburn, Simon R. / Blundo, Carlo / Burmester, Mike / Cramer, Ronald / Dawson, Ed / Gilman, Robert / Gonzalez Vasco, Maria Isabel / Grosek, Otokar / Helleseth, Tor / Kim, Kwangjo / Koblitz, Neal / Kurosawa, Kaoru / Lauter, Kristin / Lange, Tanja / Menezes, Alfred / Nguyen, Phong Q. / Pieprzyk, Josef / Rötteler, Martin / Safavi-Naini, Rei / Shparlinski, Igor E. / Stinson, Doug / Takagi, Tsuyoshi / Williams, Hugh C. / Yung, Moti

4 Issues per year


CiteScore 2016: 0.74

SCImago Journal Rank (SJR) 2016: 0.463
Source Normalized Impact per Paper (SNIP) 2016: 0.778

Mathematical Citation Quotient (MCQ) 2016: 0.16

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ISSN
1862-2984
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Volume 1, Issue 4

Issues

Minimal weight and colexicographically minimal integer representations

Clemens Heuberger / James A. Muir
  • 1 Institut für Mathematik B, Technische Universität Graz, Graz, Austria. Email:
  • 2 School of Computer Science, Carleton University, Ottawa, Canada. Email:
  • Other articles by this author:
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Published Online: 2008-02-05 | DOI: https://doi.org/10.1515/jmc.2007.015

Redundant number systems (e.g., signed binary representations) have been utilized to efficiently implement algebraic operations required by public-key cryptosystems, especially those based on elliptic curves. Several families of integer representations have been proposed that have a minimal number of nonzero digits (so-called minimal weight representations). We observe that many of the constructions for minimal weight representations actually work by building representations which are minimal in another sense. For a given set of digits, these constructions build colexicographically minimal representations; that is, they build representations where each nonzero digit is positioned as far left (toward the most significant digit) as possible. We utilize this strategy in a new algorithm which constructs a very general family of minimal weight dimension-d joint representations for any d ≥ 1. The digits we use are from the set {a ∈ Z : lau} where l ≤ 0 and u ≥ 1 are integers. By selecting particular values of l and u, it is easily seen that our algorithm generalizes many of the minimal weight representations previously described in the literature. From our algorithm, we obtain a syntactical description of a particular family of dimension-d joint representations; any representation which obeys this syntax must be both colexicographically minimal and have minimal weight; moreover, every vector of integers has exactly one representation that satisfies this syntax. We utilize this syntax in a combinatorial analysis of the weight of the representations.

Keywords: Redundant number systems; signed digits; integer representations; joint representations; minimal weight; colexicographic order; joint sparse form

About the article


Received: 2006-06-21

Published Online: 2008-02-05

Published in Print: 2007-12-01


Citation Information: Journal of Mathematical Cryptology jmc, Volume 1, Issue 4, Pages 297–328, ISSN (Online) 1862-2984, ISSN (Print) 1862-2976, DOI: https://doi.org/10.1515/jmc.2007.015.

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[1]
Clemens Heuberger
Mathematics in Computer Science, 2010, Volume 3, Number 2, Page 141
[2]
Clemens Heuberger and James A. Muir
Designs, Codes and Cryptography, 2009, Volume 52, Number 2, Page 185

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