Computing the Number of Finite Field Elements with Prescribed Trace and Co-trace

TL;DR

This paper develops a method to count finite field elements with specific trace and co-trace values by reducing the problem to solving linear equations involving Kloosterman sums, applicable to arbitrary characteristic.

Contribution

It introduces a novel approach linking the enumeration problem to linear algebra with circulant matrices derived from Kloosterman sums for any characteristic.

Findings

Method reduces counting to solving linear equations.

Applicable to arbitrary characteristic p.

Illustrated for p=2,3,5.

Abstract

In this paper, we address the problem for enumerating the number of finite field elements with prescribed trace and co-trace in case of arbitrary characteristic $p$. We show that this problem can be reduced to solving a system of $p-1$ linear equations with matrix of coefficients a slight modification of circulant matrix formed by the Kloosterman sums over the field $\MF_{p}$. The presented approach is illustrated in the cases of characteristic $p = 2,3$ and $5$.