Algorithms for Exponentiation in Finite Fields

and infinitely many integers n, multiplication in a normal basis of Fqn over Fq can be computed with O(n logn loglogn), division with O(n log n loglogn) operations in Fq , and exponentiation of an arbitrary element in Fqn with O(n2 loglogn) operations in Fq . We also prove that using a polynomial basis exponentiation in F2n can be done with the same number of operations in F2 for all n. The previous best estimates were O(n2) for multiplication in a normal basis, and O(n2 logn log logn) for exponentiation in a