Gauss periods: orders and cryptographical applications

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Gauss periods: orders and cryptographical applications

Experimental results on the multiplicative orders of Gauss periods in finite fields are presented. These results indicate that Gauss periods have high order and are often primitive (self-dual) normal elements in finite fields. It is shown that Gauss periods can be exponentiated in quadratic time. An application is an efficient pseudorandom bit generator.

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It is shown that Gauss periods of special type give an explicit polynomial-time computation of elements of exponentially large mul-tiplicative order in some nite elds. This can be considered as a step towards solving the celebrated problem of nding primitive roots in nite elds in polynomial time.

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Multiplicative Order of Gauss Periods

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.

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Order of Gauss periods in large characteristic ∗ †

Let p be the characteristic of Fq and let q be a primitive root modulo a prime r = 2n + 1. Let β ∈ Fq2n be a primitive rth root of unity. We prove that the multiplicative order of the Gauss period β + β−1 is at least (log p)c logn for some c > 0. This improves the bound obtained by Ahmadi, Shparlinski and Voloch when p is very large compared with n. We also obtain bounds for ”most” p.

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ژورنال

عنوان ژورنال: Mathematics of Computation of the American Mathematical Society

سال: 1998

ISSN: 0025-5718,1088-6842

DOI: 10.1090/s0025-5718-98-00935-1