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Normal Bases via General Gauß Periods
, 1997
"... Gau periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive rth root of unity, one obtains under certain conditions a normal basis for F q n over F q , where r is a prime and nk = r \Gamma 1 for some integer k. We generalize this constr ..."
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Gau periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive rth root of unity, one obtains under certain conditions a normal basis for F q n over F q , where r is a prime and nk = r \Gamma 1 for some integer k. We generalize this construction by allowing arbitrary integers r with nk = '(r), and find in many cases smaller values of k than is possible with the previously known approach. The first two authors are with Fachbereich17 MathematikInformatik, UniversitatGH Paderborn, D33095 Paderborn, Germany. The third author is with the International Computer Science Institutem Berkeley, USA 1 Introduction Let F q be a finite field with q elements. A basis of the form (ff; ff q ; : : : ; ff q n\Gamma1 ) of the vector space F q n over F q is a normal basis, and in this case ff is a normal element in F q n over F q . Gau periods have been used to construct normal bases in the following way: Let n; k 1 be integers ...
Computing special powers in finite fields
 e7 ← −e2 + yq; (e7 = −ypr0 + yq) 7: e8 ← −e0 + e4; (e8 = −r 2 0 + ypyq) 8: e9 ← e7e8; (e9 = (−ypr0 + yq)(−r 2 0 + ypyq)) 9: a1 ← e9 − e3 − e5; a0 ← e3 − e5 − yp; 10: a3 ← −e1 + e6; a2 ← −yp; a4 ← 0; a5 ← −yq; B Techniques for Reducing Partial Products in
, 2003
"... Abstract. We study exponentiation in nonprime finite fields with very special exponents such as they occur, for example, in inversion, primitivity tests, and polynomial factorization. Our algorithmic approach improves the corresponding exponentiation problem from about quadratic to about linear time ..."
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Abstract. We study exponentiation in nonprime finite fields with very special exponents such as they occur, for example, in inversion, primitivity tests, and polynomial factorization. Our algorithmic approach improves the corresponding exponentiation problem from about quadratic to about linear time. 1.
Exponentiation Using Addition Chains For Finite Fields
, 2000
"... We discuss two different ways to speed up exponentiation in finite fields F q n : on the one hand, reduction of the total number of operations in F q n , and on the other hand, fast computation of a single operation. Two data structures are particularly useful: sparse irreducible polynomials and nor ..."
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We discuss two different ways to speed up exponentiation in finite fields F q n : on the one hand, reduction of the total number of operations in F q n , and on the other hand, fast computation of a single operation. Two data structures are particularly useful: sparse irreducible polynomials and normal bases. We introduce weighted qaddition chains to derive efficient algorithms, and report on implementation results for our methods.
Elliptic periods for finite fields ∗
, 2008
"... We construct two new families of basis for finite field extensions. Bases in the first family, the socalled elliptic bases, are not quite normal bases, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Bases in the second family, the socalled normal ..."
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We construct two new families of basis for finite field extensions. Bases in the first family, the socalled elliptic bases, are not quite normal bases, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Bases in the second family, the socalled normal elliptic bases are normal bases and allow fast (quasilinear) arithmetic. We prove that all extensions admit models of this kind. 1