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.

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Journal Journal of Symbolic Computation
Gao, S. (Shuhong), Von Zur Gathen, J. (Joachim), Panario, D, & Shoup, V. (Victor). (2000). Algorithms for Exponentiation in Finite Fields. Journal of Symbolic Computation, 29(6), 879–889. doi:10.1006/jsco.1999.0309