High-throughput Verifiable Distributed OPRF from Gold PRF

An oblivious pseudorandom function (OPRF) is a two-party protocol that enables a client to obtain $F_k(x)$ on an input $x$ without learning the server-held key $k$, while the server learns nothing about $x$. OPRF is a fundamental building block in a wide range of privacy-preserving applications, including password-authenticated key exchange (PAKE), private set intersection (PSI), and distributed function secret sharing. In this work, we present the first concrete, high-throughput, post-quantum secure, verifiable distributed OPRF (dOPRF) tolerating $t < n/2$ malicious servers and a malicious client over replicated secret sharing, extending the recently introduced Gold OPRF of Yang et al. (IEEE S&P 2025) to a threshold setting.

We adopt an offline-online paradigm and introduce new protocol designs for both phases. (i) In the offline phase, we develop efficient protocols for batched generation of replicated secret sharing of $\alpha^e$ whose cost is independent of $e$. We present two complementary approaches, each designed for different parameter regimes: the first leverages a degenerate additive encoding under which exponentiation commutes with secret sharing; the second employs $t+1$ designated dealers that prove dual-share consistency via non-interactive zero-knowledge proofs, instantiated using both VOLE-in-the-Head and Ligero, yielding post-quantum security based solely on collision-resistant hashing. (ii) In the online phase, we propose a constant-round protocol that tightly integrates secure multiplication-and-opening with distributed zero-knowledge proof (DZKP) verification, reducing communication in every setting we evaluate and computation in all but the largest batched setting, compared to naively using the standard DZKP framework. (iii) We give a verifiable input protocol that enforces a possibly-malicious client providing well-formed inputs; its well-formedness check is absorbed into a hash exchange the online protocol already performs, and it lowers the client's upload from a full replicated share to one field element per server per input.

Our end-to-end benchmarks show that the construction substantially outperforms the state-of-the-art Legendre-PRF dOPRF of Kaluđerović et al. (ESORICS 2025) in terms of communication across all evaluated settings, and remains practical up to $(n,t)=(9,4)$, a regime where prior approaches become bandwidth- or memory-prohibitive.