On the distribution of Jacobi sums

Let \mathbf{F}_{q} be a finite field of q elements. For multiplicative characters \chi_{1},\ldots,\chi_{m} of \mathbf{F}_{q}^{\times} , we let J(\chi_{1},\ldots,\chi_{m}) denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for m=2 , the normalized Jacobi sum q^{-1/2}J(\chi_{1},\chi_{2}) ( \chi_{1}\chi_{2} nontrivial) is asymptotically equidistributed on the unit circle as q\to\infty , when \chi_{1} and \chi_{2} run through all nontrivial multiplicative characters of \mathbf{F}_{q}^{\times} . In this paper, we show a similar property for m\geq 2 . More generally, we show that the normalized Jacobi sum q^{-(m-1)/2}J(\chi_{1},\ldots,\chi_{m}) ( \chi_{1}\cdots\chi_{m} nontrivial) is asymptotically equidistributed on the unit circle, when \chi_{1},\ldots,\chi_{m} run through arbitrary sets of nontrivial multiplicative characters of \mathbf{F}_{q}^{\times} with two of the sets being sufficiently large. The case m=2 answers a question of Shparlinski.

Evaluating prime power Gauss and Jacobi sums

We show that for any mod $p^m$ characters, $\chi_1, \dots, \chi_k,$ with at least one $\chi_i$ primitive mod $p^m$, the Jacobi sum, $$ \mathop{\sum_{x_1=1}^{p^m}\dots \sum_{x_k=1}^{p^m}}_{x_1+\dots+x_k\equiv B \text{ mod } p^m}\chi_1(x_1)\cdots \chi_k(x_k), $$ has a simple evaluation when $m$ is sufficiently large (for $m\geq 2$ if $p\nmid B$). As part of the proof we give a simple evaluation of the mod $p^m$ Gauss sums when $m\geq 2$ that differs slightly from existing evaluations when $p=2$.

EULER'S CRITERION FOR SEPTIC NONRESIDUES

Let p be a prime ≡ 1 (mod 7). In this paper, we obtain an explicit expression for a primitive seventh root of unity ( mod p) in terms of coefficients of a Jacobi sum of order 7 and also in terms of a solution of a Diophantine system of Leonard and Williams, and then obtain Euler's criterion for septic nonresidues D ( mod p) in terms of this seventh root. Explicit results are given for septic nonresidues for D = 2, 3, 5, 7.

Lattice basis reduction, Jacobi sums and hyperelliptic cryptosystems

Using the LLL-algorithm for finding short vectors in lattices, we show how to compute a Jacobi sum for the prime field Fp in Q(e2πi/n) in time O(log3p), where n is small and fixed, p is large, and p = 1 (mod n). This result is useful in the construction of hyperelliptic cryptosystems.

Barnes' First Lemma and its Finite Analogue

We give a parallel proof of Barnes' first lemma and of its finite analogue. In both cases we use the Mellin transform. In the classical case, the proof avoids the residue theorem. In the finite case the Gamma function is replaced by the Gaussian sum function and the beta function by the Jacobi sum function.