A short note on Merlin-Arthur protocols for subset sum

TL;DR

This paper presents a Merlin-Arthur protocol for the subset sum problem that efficiently verifies the number of solutions with proof size and verification time proportional to the square root of the target sum, improving verification efficiency.

Contribution

It introduces a probabilistic proof system for subset sum solutions with proof size and verification time both proportional to the square root of the target sum t.

Findings

Proof size is $O^*(\sqrt{t})$

Verification time is $O^*(\sqrt{t})$

Proofs can be constructed in $O^*(t)$ time

Abstract

In the subset sum problem we are given n positive integers along with a target integer t. A solution is a subset of these integers summing to t. In this short note we show that for a given subset sum instance there is a proof of size $O^*(\sqrt{t})$ of what the number of solutions is that can be constructed in $O^*(t)$ time and can be probabilistically verified in time $O^*(\sqrt{t})$ with at most constant error probability. Here, the $O^*()$ notation omits factors polynomial in the input size $n\log(t)$.