On redundant τ-adic expansions and non-adjacent digit sets

Roberto Maria Avanzi, Clemens Heuberger, Helmut Prodinger

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


This paper studies τ-adic expansions of scalars, which are important in the design of scalar multiplication algorithms on Koblitz Curves, and are less understood than their binary counterparts. At Crypto '97 Solinas introduced the width-w τ-adic non-adjacent form for use with Koblitz curves. It is an expansion of integers z = Σi=0 z iτi, where τ is a quadratic integer depending on the curve, such that zi ≠ 0 implies zw+i-1 = ... = zi+1 = 0, like the sliding window binary recodings of integers. We show that the digit sets described by Solinas, formed by elements of minimal norm in their residue classes, are uniquely determined. However, unlike for binary representations, syntactic constraints do not necessarily imply minimality of weight. Digit sets that permit recoding of all inputs are characterized, thus extending the line of research begun by Muir and Stinson at SAC 2003 to Koblitz Curves. Two new useful digit sets are introduced: one set makes precomputations easier, the second set is suitable for low-memory applications, generalising an approach started by Avanzi, Ciet, and Sica at PKC 2004 and continued by several authors since. Results by Solinas, and by Blake, Murty, and Xu are generalized. Termination, optimality, and cryptographic applications are considered. We show how to perform a "windowed" scalar multiplication on Koblitz curves without doing precomputations first, thus reducing memory storage dependent on the base point to just one point.

Original languageEnglish
Title of host publicationSelected Areas in Cryptography - 13th International Workshop, SAC 2006, Revised Selected Papers
PublisherSpringer Verlag
Number of pages17
ISBN (Print)9783540744610
StatePublished - 2007
Externally publishedYes
Event13th International Workshop on Selected Areas in Cryptography, SAC 2006 - Montreal, QC, Canada
Duration: 17 Aug 200618 Aug 2006

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4356 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference13th International Workshop on Selected Areas in Cryptography, SAC 2006
CityMontreal, QC

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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