Verifiable geometric homomorphic secret sharing for outsourced polynomial evaluations

Homomorphic Secret Sharing (HSS) enables a client to outsource the evaluation of a function f on a private input \(x\) to multiple servers. While Shamir’s Secret Sharing (SSS)-based HSS offers notable scalability and efficiency, existing constructions predominantly operate under the semi-honest assumption. Within SSS-based architectures, existing verifiable schemes against malicious servers are highly restricted, typically limited to evaluating homogeneous polynomials over Boolean inputs. In this paper, we formalize Verifiable Information-Theoretic HSS (vIT-HSS) as a standalone primitive that provides both unconditional privacy and unconditional verifiability against malicious servers. We propose a k -server construction that is t -private and t -secure for arbitrary constant-free multivariate polynomials over any finite field. The scheme ensures that any t colluding servers learn nothing about the input and cannot cause the client to accept incorrect results, tolerating up to \(t < k\) malicious servers under the threshold condition \(( +1)k ≥ dt + 1\) , where \( \) is a tunable parameter representing the order of derivatives evaluated by the servers. Experimental results demonstrate that for medium-scale tasks, the proposed scheme reduces client-side computational overhead by over \(1,000× \) . Our work significantly broadens the verifiable evaluation capabilities of prior SSS-based HSS designs, effectively extending them to constant-free polynomials evaluated on arbitrary inputs over finite fields, providing a scalable foundation for privacy-preserving edge computing and cloud analytics.