Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lixia Luo, Changheng Li, Qiongxiu Li

TL;DR

This paper tackles the $(n,k)$-complete Hidden Subset Sum Problem (HSSP) by introducing two deterministic solvers. The first is a refined brute-force approach that leverages ordering of the unknowns to prune search, with a running time bound $O\left(\binom{n}{k}\left(\log \binom{n}{k}+\prod_{i=1}^{k-1}(\binom{n-k+i}{i}-i)\right)\right)$. The second method uses symmetric polynomials and Vieta's formulas: all elementary symmetric polynomials of the hidden multiset $X$ are derived from the $k$-subset sums, then $X$ is recovered as the roots of a univariate polynomial constructed via Vieta’s relations and Newton’s identities, yielding a bound $O\left(\sum_{u=1}^n p(u,\le k)^3+\binom{n}{k}n\right)$. A central theoretical result shows the determinant of the key transformation equals the Moser polynomial, tying invertibility to a known combinatorial quantity, and enabling exact recovery when invertible. The work also explores homogeneous symmetric polynomial rings, highlighting deeper algebraic structure and potential AI-privacy applications.

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.

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

TL;DR

This paper tackles the -complete Hidden Subset Sum Problem (HSSP) by introducing two deterministic solvers. The first is a refined brute-force approach that leverages ordering of the unknowns to prune search, with a running time bound . The second method uses symmetric polynomials and Vieta's formulas: all elementary symmetric polynomials of the hidden multiset are derived from the -subset sums, then is recovered as the roots of a univariate polynomial constructed via Vieta’s relations and Newton’s identities, yielding a bound . A central theoretical result shows the determinant of the key transformation equals the Moser polynomial, tying invertibility to a known combinatorial quantity, and enabling exact recovery when invertible. The work also explores homogeneous symmetric polynomial rings, highlighting deeper algebraic structure and potential AI-privacy applications.

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 -complete HSSP, where a target multiset must be recovered from its all -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 complexity, where counts the number of partitions of a positive integer into at most parts. The latter constructs an -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.

Paper Structure

This paper contains 15 sections, 8 theorems, 52 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Brute-force search
  4. The approach based on Vieta's formulas
  5. Some notations and the preliminary idea
  6. Constructing the polynomial we needed
  7. Computing $P_1$
  8. Computing $P_2,P_{(1,1)}$
  9. Computing $P_{p}$ for any partition $p$ of $u$
  10. Applying Newton's identities and Vieta's formulas
  11. The determinant of Matrix \ref{['eq:mat']}
  12. Symmetric subset sum recovery algorithm
  13. Nontrivial permutation-symmetric subset of $X_{n,k}$
  14. The homogeneous symmetric polynomial rings
  15. Conclusion and future work

Key Result

Theorem 1

There is a deterministic algorithm to solve the $(n,k)$-complete HSSP defined over $\mathbb{R}$ via brute-force search, with a time complexity of

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 1
  • proof
  • Theorem 3
  • ...and 13 more