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Gauss Sums of Cubic Characters over Fpr, p Odd

Schipani D., Elia M.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

Vol. 60, no 1

1--19

An elementary approach is shown which derives the values of the Gauss sums over Fpr, p odd, of a cubic character. New links between Gauss sums over different field extensions are shown in terms of factorizations of the Gauss sums themselves, which are then revisited in terms of prime ideal decompositions. Interestingly, one of these results gives a representation of primes p of the form 6k+1 by a binary quadratic form in integers of a subfield of the cyclotomic field of the pth roots of unity.

Gauss sums of the cubic character over GF(2m): an elementary derivation

Schipani D., Elia M.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

Vol. 59, no 1

11-18

By an elementary approach, we derive the value of the Gauss sum of a cubic character over a finite field F2s without using Davenport-Hasse's theorem (namely, if s is odd the Gauss sum is -1, and if s is even its value is -(-2)s/2).

Primality proving with Gauss and Jacobi sums

Chmielowiec A.

Journal of Telecommunications and Information Technology

2004

nr 4

69-75

This article presents a primality test known as APR (Adleman, Pomerance and Rumely) which was invented in 1980. It was later simplified and improved by Cohen and Lenstra. It can be used to prove primality of numbers with thousands of bits in a reasonable amount of time. The running time of this algorithm for number N is O((lnN)Cln ln lnN) for some constant C. This is almost polynomial time since for all practical purposes the function ln ln lnN acts like a constant.