Scalar multiplication on Koblitz curves using double bases

Roberto Avanzi, Francesco Sica

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


The paper is an examination of double-base decompositions of integers n, namely expansions loosely of the form n = Σi, j ± A iBj for some base {A, B}. This was examined in previous works [5,6], in the case when A, B lie in N. We show here how to extend the results of [5] to Koblitz curves over binary fields. Namely, we obtain a sublinear scalar algorithm to compute, given a generic positive integer n and an elliptic curve point P, the point nP in time O (log n / log lgo n) elliptic curve operations with essentially no storage, thus making the method asymptotically faster than any know scalar multiplication algorithm on Koblitz curves. In view of combinatorial results, this is the best type of estimate with two bases, apart from the value of the constant in the O notation.

Original languageEnglish
Title of host publicationProgress in Cryptology, VIETCRYPT 2006 - 1st International Conference on Cryptology in Vietnam, Revised Selected Papers
PublisherSpringer Verlag
Number of pages16
ISBN (Print)3540687998, 9783540687993
StatePublished - 2006
Externally publishedYes
Event1st International Conference on Cryptology in Vietnam, VIETCRYPT 2006 - Hanoi, Viet Nam
Duration: 25 Sep 200628 Sep 2006

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4341 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference1st International Conference on Cryptology in Vietnam, VIETCRYPT 2006
Country/TerritoryViet Nam


  • Double base number systems
  • Elliptic curves
  • Koblitz curves
  • Scalar multiplication
  • Sublinear algorithms

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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