Extending scalar multiplication using double bases

Roberto Avanzi, Vassil Dimitrov, Christophe Doche, Francesco Sica

Research output: Chapter in Book/Report/Conference proceedingConference contribution

33 Citations (Scopus)

Abstract

It has been recently acknowledged [4,6,9] that the use of double bases representations of scalars n, that is an expression of the form n = ∑e,s,t (-1)e As Bt can speed up significantly scalar multiplication on those elliptic curves where multiplication by one base (say B) is fast. This is the case in particular of Koblitz curves and supersingular curves, where scalar multiplication can now be achieved in o(logn) curve additions. Previous literature dealt basically with supersingular curves (in characteristic 3, although the methods can be easily extended to arbitrary characteristic), where A,B ∈ ℕ. Only [4] attempted to provide a similar method for Koblitz curves, where at least one base must be non-real, although their method does not seem practical for cryptographic sizes (it is only asymptotic), since the constants involved are too large. We provide here a unifying theory by proposing an alternate recoding algorithm which works in all cases with optimal constants. Furthermore, it can also solve the until now untreatable case where both A and B are non-real. The resulting scalar multiplication method is then compared to standard methods for Koblitz curves. It runs in less than logn/loglogn elliptic curve additions, and is faster than any given method with similar storage requirements already on the curve K-163, with larger improvements as the size of the curve increases, surpassing 50% with respect to the τ-NAF for the curves K-409 and K-571. With respect of windowed methods, that can approach our speed but require O(log(n)/loglog(n)) precomputations for optimal parameters, we offer the advantage of a fixed, small memory footprint, as we need storage for at most two additional points.

Original languageEnglish
Title of host publicationAdvances in Cryptology - ASIACRYPT 2006 - 12th International Conference on the Theory and Application of Cryptology and Information Security, Proceedings
Pages130-144
Number of pages15
DOIs
Publication statusPublished - Dec 1 2006
Event12th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2006 - Shanghai, China
Duration: Dec 3 2006Dec 7 2006

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4284 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other12th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2006
CountryChina
CityShanghai
Period12/3/0612/7/06

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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    Avanzi, R., Dimitrov, V., Doche, C., & Sica, F. (2006). Extending scalar multiplication using double bases. In Advances in Cryptology - ASIACRYPT 2006 - 12th International Conference on the Theory and Application of Cryptology and Information Security, Proceedings (pp. 130-144). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4284 LNCS). https://doi.org/10.1007/11935230_9