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Gauß periods in finite fields. (English) Zbl 1019.11036

Jungnickel, Dieter (ed.) et al., Finite fields and applications. Proceedings of the fifth international conference on finite fields and applications \(F_q5\), University of Augsburg, Germany, August 2-6, 1999. Berlin: Springer. 162-177 (2001).
The paper under review is a survey article on Gauss periods. Although originally defined (by Gauss) over the rational field, the analogous definition makes sense over any finite field \(\mathbb F_q\). Let \(r\) be a prime with \(\gcd(q,r)=1\), and let \(\beta\) be a primitive \(r\)-th root of unity in \(\mathbb F_{q^{r - 1}}\). Let \(n, k\) be positive integers with \(nk = r - 1\), and let \(K\) be the (unique) cyclic subgroup of order \(k\) in the multiplicative group of \(\mathbb F_r\). Then \(\alpha = \sum_{i\in K} \beta^i \in \mathbb F_{q^n}\) is called a Gauss period of type \((n, K)\) over \(\mathbb F_q\).
This review concentrates on two applications of these Gauss periods in finite fields. The first involves finding elements of exponentially large order in certain finite fields. One can think of this as a potential beginning step towards finding an efficient algorithm for constructing a primitive element in a given finite field.
The second application deals with efficient exponentiation in finite fields. Gauss periods lead to the fastest algorithms for this problem, an important component of certain cryptographic systems.
For the entire collection see [Zbl 0959.00027].

MSC:

11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)
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