TY - JOUR

T1 - Algorithms for Exponentiation in Finite Fields

AU - Gao, Shuhong

AU - Von Zur Gathen, Joachim

AU - Panario, Daniel

AU - Shoup, Victor

N1 - Copyright:
Copyright 2004 Elsevier Science B.V., Amsterdam. All rights reserved.

PY - 2000/6

Y1 - 2000/6

N2 - 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.

AB - 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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U2 - 10.1006/jsco.1999.0309

DO - 10.1006/jsco.1999.0309

M3 - Article

AN - SCOPUS:0000185837

VL - 29

SP - 879

EP - 889

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

SN - 0747-7171

IS - 6

ER -