Width-3 joint sparse form

Katsuyuki Okeya, Hidehiro Kato, Yasuyuki Nogami

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

4 Citations (Scopus)


The joint sparse form (JSF) is a representation of a pair of integers, which is famous for accelerating a multi-scalar multiplication in elliptic curve cryptosystems. Solinas' original paper showed three unsolved problems on the enhancement of JSF. Whereas two of them have been solved, the other still remains to be done. The remaining unsolved problem is as follows: To design a representation of a pair of integers using a larger digit set such as a set involving ±3, while the original JSF utilizes the digit set that consists of 0,±1 for representing a pair of integers. This paper puts an end to the problem; width-3 JSF. The proposed enhancement satisfies properties that are similar to that of the original. For example, the enhanced representation is defined as a representation that satisfies some rules. Some other properties are the existence, the uniqueness of such a representation, and the optimality of the Hamming weight. The non-zero density of the width-3 JSF is 563/1574(= 0.3577) and this is ideal. The conversion algorithm to the enhanced representation takes O(log n) memory and O(n) computational cost, which is very efficient, where n stands for the bit length of the integers. Keywords: elliptic curve cryptosystem, multi-scalar multiplication, joint sparse form (JSF).

Original languageEnglish
Title of host publicationInformation Security Practice and Experience - 6th International Conference, ISPEC 2010, Proceedings
Number of pages18
Publication statusPublished - Dec 23 2010
Event6th International Conference on Information Security Practice and Experience, ISPEC 2010 - Seoul, Korea, Republic of
Duration: May 12 2010May 13 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6047 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other6th International Conference on Information Security Practice and Experience, ISPEC 2010
Country/TerritoryKorea, Republic of


  • Joint sparse form (JSF)
  • Ulliptic curve cryptosystem
  • Uulti-scalar multiplication

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


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