Improved algorithms for efficient arithmetic on elliptic curves using fast endomorphisms

Mathieu Ciet, Tanja Lange, Francesco Sica, Jean Jacques Quisquater

Research output: Contribution to journalArticle

22 Citations (Scopus)


In most algorithms involving elliptic curves, the most expensive part consists in computing multiples of points. This paper investigates how to extend the τ-adic expansion from Koblitz curves to a larger class of curves defined over a prime field having an efficiently-computable endomorphism φ in order to perform an efficient point multiplication with efficiency similar to Solinas' approach presented at CRYPTO '97. Furthermore, many elliptic curve cryptosystems require the computation of k0P + k1Q. Following the work of Solinas on the Joint Sparse Form, we introduce the notion of φ-Joint Sparse Form which combines the advantages of a φ-expansion with the additional speedup of the Joint Sparse Form. We also present an efficient algorithm to obtain the φ-Joint Sparse Form. Then, the double exponentiation can be done using the φ endomorphism instead of doubling, resulting in an average of l applications of φ and l/2 additions, where l is the size of the ki's. This results in an important speed-up when the computation of φ is particularly effective, as in the case of Koblitz curves.

Original languageEnglish
Pages (from-to)388-400
Number of pages13
JournalLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Publication statusPublished - Dec 1 2003



  • Elliptic curves
  • Fast endomorphisms
  • Joint Sparse Form

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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