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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

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CiteScore 2016: 0.74

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Source Normalized Impact per Paper (SNIP) 2016: 0.778

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Volume 2, Issue 4

Minimal weight expansions in Pisot bases

Christiane Frougny / Wolfgang Steiner
  • LIAFA, CNRS UMR 7089, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France. Email:
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Published Online: 2009-02-04 | DOI: https://doi.org/10.1515/JMC.2008.017


For applications to cryptography, it is important to represent numbers with a small number of non-zero digits (Hamming weight) or with small absolute sum of digits. The problem of finding representations with minimal weight has been solved for integer bases, e.g. by the non-adjacent form in base 2. In this paper, we consider numeration systems with respect to real bases β which are Pisot numbers and prove that the expansions with minimal absolute sum of digits are recognizable by finite automata. When β is the Golden Ratio, the Tribonacci number or the smallest Pisot number, we determine expansions with minimal number of digits ±1 and give explicitely the finite automata recognizing all these expansions. The average weight is lower than for the non-adjacent form.

Keywords.: Minimal weight; beta-expansions; Pisot numbers; Fibonacci numbers; automata

About the article

Received: 2008-03-27

Revised: 2008-12-16

Published Online: 2009-02-04

Published in Print: 2008-12-01

Citation Information: Journal of Mathematical Cryptology, Volume 2, Issue 4, Pages 365–392, ISSN (Online) 1862-2984, ISSN (Print) 1862-2976, DOI: https://doi.org/10.1515/JMC.2008.017.

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