Koblitz curves and integer equivalents of frobenius expansions

Billy Bob Brumley, Kimmo Järvinen

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

9 Citations (Scopus)


Scalar multiplication on Koblitz curves can be very efficient due to the elimination of point doublings. Modular reduction of scalars is commonly performed to reduce the length of expansions, and τ-adic Non-Adjacent Form (NAF) can be used to reduce the density. However, such modular reduction can be costly. An alternative to this approach is to use a random τ-adic NAF, but some cryptosystems (e.g. ECDSA) require both the integer and the scalar multiple. This paper presents an efficient method for computing integer equivalents of random τ-adic expansions. The hardware implications are explored, and an efficient hardware implementation is presented. The results suggest significant computational efficiency gains over previously documented methods.

Original languageEnglish
Title of host publicationSelected Areas in Cryptography - 14th International Workshop, SAC 2007, Revised Selected Papers
Number of pages12
Publication statusPublished - 1 Dec 2007
Externally publishedYes
Publication typeA4 Article in conference proceedings
Event14th International Workshop on Selected Areas in Cryptography, SAC 2007 - Ottawa, Canada
Duration: 16 Aug 200717 Aug 2007

Publication series

NameLecture Notes in Computer Science
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference14th International Workshop on Selected Areas in Cryptography, SAC 2007


  • Digital signatures
  • Elliptic curve cryptography
  • Koblitz curves

Publication forum classification

  • Publication forum level 1

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


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