Christophe Doche*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

5 Citations (Scopus)


Given an element x of a group (G,×) and an integer n ∈ Z one describes in this chapter efficient methods to perform the exponentiation xn. Only positive exponents are considered since xn = (1/x)−n but nothing more is assumed especially regarding the structure and the properties of G. See Chapter 11 for specific improvements concerning finite fields. Two elementary operations are used, namely multiplications and squarings. The distinction is made for performance reasons since squarings can often be implemented more efficiently; see Chapters 10 and 11 for details. In the context of elliptic and hyperelliptic curves, the computations are done in an abelian group denoted additively (G,⊕). The equivalent of the exponentiation xn is the scalar multiplication [n]P. All the techniques described in this chapter can be adapted in a trivial way, replacing multiplication by addition and squaring by doubling.

Original languageEnglish
Title of host publicationHandbook of elliptic and hyperelliptic curve cryptography
Place of PublicationBoca Raton, Florida, USA
PublisherCRC Press, Taylor & Francis Group
Number of pages24
ISBN (Electronic)9781420034981
ISBN (Print)9781584885184
Publication statusPublished - 2006

Publication series

NameDiscrete mathematics and its applications
PublisherChapman & Hall/CRC


  • exponentiation
  • exponent
  • base point


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