MULTIPLICATIVE ORDER OF GAUSS PERIODS
نویسندگان
چکیده
منابع مشابه
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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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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H. Hasse conjectured that all multiplicative relations between Gauss sums essentially follow from the Davenport-Hasse product formula and the norm relation for Gauss sums. While this is known to be false, very few counterexamples, now known as sign ambiguities, have been given. Here, we provide an explicit product formula giving an infinite class of new sign ambiguities and resolve the ambiguou...
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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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A result on finite abelian groups is first proved and then used to solve problems in finite fields. Particularly, all finite fields that have normal bases generated by general Gauss periods are characterized and it is shown how to find normal bases of low complexity. Dedicated to Professor Chao Ko on his 90th birthday.
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2010
ISSN: 1793-0421,1793-7310
DOI: 10.1142/s1793042110003290