Efficient Pseudorandom Correlation Generators over $\mathbb {Z}/p^k\mathbb {Z}$

Modern efficient secure multi-party computation (MPC) protocols typically follow an offline-online design, where offline protocols produce a sufficient amount of correlated randomness that would be consumed during the online phases. The past decades have witnessed maturing of efficient online protocols, for computing circuits over either arbitrary finite fields or rings $\mathbb{Z}_{p^k}$. In particular, protocols tailored for $\mathbb{Z}_{2^k}$ arithmetic have achieved better concrete efficiency in most real-life applications, as it naturally captures modern CPU architectures. On the other hand, a recent paradigm of pseudorandom correlation generator (PCG) initiated by Boyle et al. (CCS'18, Crypto'19) opens a door to efficient preprocessing with sublinear communication. Since then, PCGs have been extensively studied and developed to produce various types of correlations required from online protocols. Although Li et al. (EuroCrypt'25) recently put a significant step forward and propose efficient PCGs for arbitrary finite fields, the current state of PCGs for rings is not satisfying at all. Towards the great demand for efficiently generating correlations over rings, we investigate PCGs for general Galois rings, which simultaneously unify finite fields and integer rings modulo $p^k$. In summary, we establish the following results:

(i) We generalize the state-of-the-art PCG constructions for oblivious linear evaluations (OLE) over Galois fields to arbitrary Galois rings, basing on Galois theory and the Hensel lift. Moreover, our PCGs for Galois rings are as efficient as PCGs for fields. Concretely, for $mN$ OLE correlations over $\mathbb{Z}_{2^k}$, we require $O(m\log{N})$ communication and $O(m^2N\log{N})$ computation, where $m$ is an arbitrary integer $\geq 2$. In comparison, to our best knowledge, previous approaches incur communication at least linear in $N$.

(ii) We extend the above OLE construction to provide various types of correlations over any Galois ring. One of the fascinating applications is an efficient PCG for two-party SPD$\mathbb{Z}_{2^k}$ authenticated multiplication triples (Crypto'18). For $mN$ SPD$\mathbb{Z}_{2^k}$ triples, our approach requires only $O(m\log{N})$ communication and $O(m^2N\log{N})$ computation. Concrete evaluations show that our method significantly outperforms existing schemes based on homomorphic encryption.

(iii) In addition, our PCGs for Galois rings also enable multi-party multiplication triple generation, yielding the first efficient MPC protocol for arithmetic circuits over $\mathbb{Z}_{2^k}$ with silent and sublinear preprocessing. Additional applications include circuit-dependent preprocessing and matrix multiplication triples, etc, which are of independent interest.