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

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.