Moment subset sums over finite fields

TL;DR

This paper introduces a deterministic polynomial-time algorithm for the higher moment subset sum problem over finite fields, applicable to specific polynomial image sets, extending classical results for the case m=1.

Contribution

It provides the first polynomial-time solution for the m-th moment subset sum problem over finite fields for fixed m and specific polynomial image sets.

Findings

Polynomial-time algorithm for fixed m and specific polynomial images

Extends classical subset sum results to higher moments

Applicable to monomial and Dickson polynomial image sets

Abstract

The $k$-subset sum problem over finite fields is a classical NP-complete problem.Motivated by coding theory applications, a more complex problem is the higher $m$-th moment $k$-subset sum problem over finite fields. We show that there is a deterministic polynomial time algorithm for the $m$-th moment $k$-subset sum problem over finite fields for each fixed $m$ when the evaluation set is the image set of a monomial or Dickson polynomial of any degree $n$. In the classical case $m=1$, this recovers previous findings.