About cookies on this site Our websites require some cookies to function properly (required). In addition, other cookies may be used with your consent to analyze site usage, improve the user experience and for advertising. For more information, please review your options. By visiting our website, you agree to our processing of information as described in IBM’sprivacy statement. To provide a smooth navigation, your cookie preferences will be shared across the IBM web domains listed here.
Publication
Journal of Symbolic Computation
Paper
Algorithms for Exponentiation in Finite Fields
Abstract
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. © 2000 Academic Press.