Jump to ContentJump to Main Navigation
Show Summary Details
More options …

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

See all formats and pricing
More options …
Volume 1, Issue 4


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:
  • De Gruyter OnlineGoogle Scholar
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.

Export Citation

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

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

Comments (0)

Please log in or register to comment.
Log in