# Computation of Jacobi sums of order l^2 and 2l^2 with prime l

## TL;DR

This paper introduces efficient algorithms for computing Jacobi sums of orders l^2 and 2l^2 with prime l, utilizing cyclotomic numbers, and validates these methods through implementation.

## Contribution

It provides new fast algorithms for Jacobi sums of specific orders and demonstrates their minimal cyclotomic number requirements.

## Key findings

- Algorithms significantly reduce computation time for Jacobi sums.
- Validation confirms minimal cyclotomic numbers are sufficient.
- Implementation showcases practical applicability of the formulas.

## Abstract

In this paper, we present the fast computational algorithms for the Jacobi sums of orders $l^2$ and $2l^{2}$ with odd prime $l$ by formulating them in terms of the minimum number of cyclotomic numbers of the corresponding orders. We also implement two additional algorithms to validate these formulae, which are also useful for the demonstration of the minimality of cyclotomic numbers required.