Finite field arithmetic

Christophe Doche*

*Corresponding author for this work

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

5 Citations (Scopus)


In this chapter, we are mainly interested in performance; see Section 2.3 for a theoretical presentation of finite fields. In the following, we consider three kinds of fields that are of great cryptographic importance, namely prime fields, extension fields of characteristic 2, and optimal extension fields. We will describe efficient methods for performing elementary operations, such as addition, multiplication, inversion, exponentiation, and square roots. The material that we give is implicitly more related to a software approach; see Chapter 26 for a presentation focused on hardware. Efficient finite field arithmetic is crucial in efficient elliptic or hyperelliptic curve cryptosystems and is the subject of abundant literature [JUN 1993, LINI 1997, SHP 1999]. See also the preliminary version of a book written by Shoup and available online [SHO], introducing basic concepts from computational number theory and algebra, and including all the necessary mathematical background.

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 pages37
ISBN (Electronic)9781420034981
ISBN (Print)9781584885184
Publication statusPublished - 2006

Publication series

NameDiscrete mathematics and its applications
PublisherChapman & Hall/CRC


  • finite fields
  • prime fields
  • extension fields of characteristic 2
  • optimal extension fields


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