An efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings is presented. Also a quantum algorithm for the discrete logarithm problem is described, which is different from Shor’s algorithm. A reduction from the discrete logarithm problem to Gauss sum estimation gives evidence that the latter is classically a hard problem. A crucial ingredient of the algorithms i...

In this paper, we use the elementary methods, properties of Dirichlet character sums and classical Gauss to study estimation mean value high-powers for a special sum modulo prime, derive an exact computational formula. It can be conveniently programmed by “Mathematica” software, which get results easily.

Recently, linear codes with a few weights were widely investigated due to their applications in secret sharing schemes and authentication schemes. In this letter, we present a class of q-ary linear codes derived from irreducible cyclic codes with q a prime power. We use Gauss sums to represent its Hamming weights and obtain the bounds of its minimum Hamming distance. In some cases, we explicitl...

The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs, building upon earlier work of C. André. We study supercharacter theories on (Z/nZ)d induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, and Kloosterman sums) arise ...

We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and Gauss are used to obtain these formulas. These (along with 2-adic Siegel’s mass formula) can be compute the representation numbers certain forms.

Evaluating sums of multivariate Gaussians is a common computational task in computer vision and pattern recognition, including in the general and powerful kernel density estimation technique. The quadratic computational complexity of the summation is a significant barrier to the scalability of this algorithm to practical applications. The fast Gauss transform (FGT) has successfully accelerated ...