# Polynomial Criterion for Abelian Difference Sets

## TL;DR

This paper introduces a polynomial criterion for identifying abelian difference sets by linking them to solutions of polynomial equations, and explores implications for testing Boolean bent functions.

## Contribution

It establishes a novel correspondence between abelian difference sets and solutions to polynomial systems, providing new tools for their analysis and for Boolean function testing.

## Key findings

- Characterizes difference sets via polynomial solutions
- Provides tests for Boolean bent functions
- Links algebraic structures with combinatorial properties

## Abstract

Difference sets are subsets of a group satisfying certain combinatorial property with respect to the group operation. They can be characterized using an equality in the group ring of the corresponding group. In this paper, we exploit the special structure of the group ring of an abelian group to establish a one-to one correspondence of the class of difference sets with specific parameters in that group with the set of all complex solutions of a specified system of polynomial equations. The correspondence also develops some tests for a Boolean function to be a bent function.