Algorithms for Exponentiation in Finite Fields

Shuhong Gao, Joachim Von Zur Gathen, Daniel Panario, Victor Shoup

Research output: Contribution to journalArticlepeer-review


Gauss periods yield (self-dual) normal bases in finite fields, and these normal bases can be used to implement arithmetic efficiently. It is shown that for a small prime power q and infinitely many integersn , multiplication in a normal basis of Fqn over Fq can be computed with O(n logn loglog n), division with O(n log2n loglog n) operations in Fq, and exponentiation of an arbitrary element in Fqn withO (n2loglog n) operations in Fq. We also prove that using a polynomial basis exponentiation in F 2 n can be done with the same number of operations in F 2 for all n. The previous best estimates were O(n2) for multiplication in a normal basis, andO (n2log n loglog n) for exponentiation in a polynomial basis.

Original languageEnglish (US)
Pages (from-to)879-889
Number of pages11
JournalJournal of Symbolic Computation
Issue number6
StatePublished - Jun 2000

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics


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