Deterministic Algorithms to Solve the $(n,k)$-Complete Hidden Subset Sum Problem

TL;DR

This paper introduces two deterministic algorithms for the $(n,k)$-complete Hidden Subset Sum Problem, including a novel polynomial-based method that improves computational efficiency using symmetric polynomials and Vieta's formulas.

Contribution

The paper presents a new polynomial-based deterministic algorithm for the $(n,k)$-complete HSSP, utilizing symmetric polynomials and Vieta's formulas, with a detailed complexity analysis.

Findings

The polynomial-based algorithm constructs an $n$-th degree polynomial with roots as the hidden elements.

The complexity of the new method is $O( extstyleig( extstyle ext{sum}_{u=1}^n p(u, ext{≤}k)^3 + inom{n}{k}nig))$.

Homogeneous symmetric polynomial rings are discussed as an independent theoretical contribution.

Abstract

The Hidden Subset Sum Problem (HSSP) is a significant NP-complete problem in number theory and combinatorics, with applications in cryptography and AI privacy. For the $(n,k)$-complete HSSP, where a target multiset must be recovered from its all $k$-subset sums, existing algorithms face limitations due to high complexity or intractability. This paper proposes two deterministic algorithms: a brute-force approach, and a novel method leveraging symmetric polynomials and Vieta's formulas with $O\left(\sum_{u=1}^n p(u,\leq k)^3+\binom{n}{k}n\right)$ complexity, where $ p(u,\leq k)$ counts the number of partitions of a positive integer $u$ into at most $k$ parts. The latter constructs an $n$-th degree polynomial via Vieta's formulas, whose roots correspond to the hidden multiset elements. Additionally, the discussion about the homogeneous symmetric polynomial rings is of independent interest.