Paper 2026/104 Original: eprint.iacr.org/2026/104 ↗

Deal out oblivious correlations: 2-depth HSS circuit for silent V-OLE generation

Abstract

We analyzed in depth the Homomorphic Secret Sharing construction applied for Pseudorandom Correlation Function, and we obtained interesting results for various applications. In this paper, we discuss how the PCF can be achieved using the Damgard-Jurik HSS schema by solving the distance function over a ciphertext parametric space of \(\mathbb{Z}^{*}_{n^{\zeta + 1}}\), performing the distributed multiplication protocol as the base building block for our PCF. We created a weak PCF for Vector-OLE via 1-depth HSS circuit, furthermore, via what we called pre-computation with RO-less, we achieved a strong PCF for V-OLE between two parties correct against anhonest-but-curious adversary \(\mathcal{A}_{\mathsf{hbc}}\) and fail-safe secure against an active adversary \(\mathcal{A}_{\mathsf{poly}}\). We also extended our main construction by describing a silent approach in two different ways described as semi-silent by a pre-sampling assumption between the parties and a true-silent protocol execution exploiting the generation of seeds by a PRF. As a last step, we discussed how to build a \(n \times\)OLE generator via our pre-computation session to craft an arbitrary amount of OLE correlation. Our entire paper is further verified by the implementation of a complete and exhaustive library covering all HSS operations, publicly accessible and usable via \(\textit{pip install obliviouspy-HSS}\) and importing \(\textit{import oblivious}\).