International Association
for Cryptologic Research

We revisit the alternating moduli paradigm for constructing symmetric key primitives with a focus on constructing highly efficient protocols to evaluate them using secure multi-party computation (MPC). The alternating moduli paradigm of Boneh et al. (TCC 2018) enables the construction of various symmetric key primitives with the common characteristic that the inputs are multiplied by two linear maps over different moduli, first over F_2 and then over F_3. The first contribution focuses on efficient two-party evaluation of alternating moduli PRFs, effectively building an oblivious pseudorandom function. We present a generalization of the PRF proposed by Boneh et al. (TCC 18) along with methods to lower the communication and computation. We then provide several variants of our protocols, with different computation and communication tradeoffs, for evaluating the PRF. Most are in the OT/VOLE hybrid model while one is based on specialized garbling. Our most efficient protocol effectively is about 3x faster and requires 1.3x lesser communication. Our next contribution is the efficient evaluation of the OWF f(x) = B *_3 (A *_2 x) proposed by Dinur et al. (CRYPTO 21) where A \in F^{m x n}_2, B \in F^{t x m}_3 and *_p is multiplication mod p. This surprisingly simple OWF can be evaluated within MPC by secret sharing [x] over F_2, locally computing [v] = A *_2 [x], performing a modulus switching protocol to F_3 shares, followed by locally computing the output shares [y] = B *_3 [v]. We design a bespoke MPC-in-the-Head (MPCitH) signature scheme that evaluates the OWF, achieving state of art performance. The resulting signature has a size ranging from 4.0-5.5 KB, achieving between 2-3x reduction compared to Dinur et al. To the best of our knowledge, this is only 5% larger than the smallest signature based on symmetric key primitives, including the latest NIST PQC competition submissions. We additionally show that our core techniques can be extended to build very small post-quantum ring signatures for small-medium sized rings that are competitive with state-of-the-art lattice based schemes. Our techniques are in fact more generally applicable to set membership in MPCitH.

The recent development of pseudorandom correlation generators (PCG) holds tremendous promise for highly efficient MPC protocols. Among other correlations, PCGs allow for the efficient generation of oblivious transfer (OT) and vector oblivious linear evaluations (VOLE) with sublinear communication and concretely good computational overhead. This type of PCG makes use of a so-called LPN-friendly error-correcting code. That is, for large dimensions the code should have very efficient encoding and have high minimum distance. We investigate existing LPN-friendly codes and find that several candidates are less secure than was believed. Beginning with the recent expand-accumulate codes, we find that for their aggressive parameters, aimed at good concrete efficiency, they achieve a smaller minimum distance than conjectured. This decreases the resulting security parameter of the PCG but it remains unclear by how much. We additionally show that the recently proposed and extremely efficient silver codes achieve only very small minimum distance and result in concretely efficient attacks on the resulting PCG protocol. As such, silver codes should not be used. We introduce a new LPN-friendly code which we call expand-convolute. These codes have provably high minimum distance and faster encoding time than suitable alternatives, e.g. expand-accumulate. The main contribution of these codes is the introduction of a convolution step that dramatically increases the minimum distance. This in turn allows for a more efficient parameter selection which results in improved concrete performance. In particular, we observe a 2 times improvement in running time.

We study the problem of verifiably outsourcing computation in a model where the verifier has black-box access to the function being computed. We introduce the problem of oracle-aided batch verification of computation (OBVC) for a function class F. This allows a verifier to efficiently verify the correctness of any f \in F evaluated on a batch of n instances x_1, ...., x_n, while only making \lambda calls to an oracle for f (along with O(n \lambda) calls to low-complexity helper oracles), where \lambda denotes a security parameter. We obtain the following positive and negative results: 1. We build OBVC protocols for the class F of all functions that admit random-self-reductions. Some of our protocols rely on homomorphic encryption schemes. 2. We show that there cannot exist OBVC schemes for the class F of all functions mapping \lambda-bit inputs to \lambda-bit outputs, for any n = \poly(\lambda).

In this work we present a new construction for a batched Oblivious Pseudorandom Function (OPRF) based on Vector-OLE and the PaXoS data structure. We then use it in the standard transformation for achieving Private Set Intersection (PSI) from an OPRF. Our overall construction is highly efficient with $O(n)$ communication and computation. We demonstrate that our protocol can achieve malicious security at only a very small overhead compared to the semi-honest variant. For input sizes $n = 2^{20}$, our malicious protocol needs 6.2 seconds and less than 59 MB communication. This corresponds to under 450 bits per element, which is the lowest number for any published PSI protocol (semi-honest or malicious) to date. Moreover, in theory our semi-honest (resp. malicious) protocol can achieve as low as 219 (resp. 260) bits per element for $n=2^{20}$ at the added cost of interpolating a polynomial over $n$ elements. As a second contribution, we present an extension where the output of the PSI is secret-shared between the two parties. This functionality is generally referred to as Circuit-PSI. It allows the parties to perform a subsequent MPC protocol on the secret-shared outputs, e.g., train a machine learning model. Our circuit PSI protocol builds on our OPRF construction along with another application of the PaXoS data structure. It achieves semi-honest security and allows for a highly efficient implementation, up to 3x faster than previous work.

In multi-party threshold private set intersection (PSI), $n$ parties each with a private set wish to compute the intersection of their sets if the intersection is sufficiently large. Previously, Ghosh and Simkin (CRYPTO 2019) studied this problem for the two-party case and demonstrated interesting lower and upper bounds on the communication complexity. In this work, we investigate the communication complexity of the multi-party setting $(n\geq 2)$. We consider two functionalities for multi-party threshold PSI. In the first, parties learn the intersection if each of their sets and the intersection differ by at most $T$. In the second functionality, parties learn the intersection if the union of all their sets and the intersection differ by at most $T$. For both functionalities, we show that any protocol must have communication complexity $\Omega(nT)$. We build protocols with a matching upper bound of $O(nT)$ communication complexity for both functionalities assuming threshold FHE. We also construct a computationally more efficient protocol for the second functionality with communication complexity $\widetilde{O}(nT)$ under a weaker assumption of threshold additive homomorphic encryption. As a direct implication, we solve one of the open problems in the work of Ghosh and Simkin (CRYPTO 2019) by designing a two-party protocol with communication cost $\widetilde{O}(T)$ from assumptions weaker than FHE. As a consequence of our results, we achieve the first "regular" multi-party PSI protocol where the communication complexity only grows with the size of the set difference and does not depend on the size of the input sets.

We put forth new protocols for oblivious transfer extension and vector OLE, called \emph{Silver}, for SILent Vole and oblivious transfER. Silver offers extremely high performances: generating 10 million random OTs on one core of a standard laptop requires only 300ms of computation and 122KB of communication. This represents 37% less computation and ~1300x less communication than the standard IKNP protocol, as well as ~4x less computation and ~4x less communication than the recent protocol of Yang et al. (CCS 2020). Silver is \emph{silent}: after a one-time cheap interaction, two parties can store small seeds, from which they can later \emph{locally} generate a large number of OTs \emph{while remaining offline}. Neither IKNP nor Yang et al. enjoys this feature; compared to the best known silent OT extension protocol of Boyle et al. (CCS 2019), upon which we build up, Silver has 19x less computation, and the same communication. Due to its attractive efficiency features, Silver yields major efficiency improvements in numerous MPC protocols. Our approach is a radical departure from the standard paradigm for building MPC protocols, in that we do \emph{not} attempt to base our constructions on a well-studied assumption. Rather, we follow an approach closer in spirit to the standard paradigm in the design of symmetric primitives: we identify a set of fundamental structural properties that allow us to withstand all known attacks, and put forth a candidate design, guided by our analysis. We also rely on extensive experimentations to analyze our candidate and experimentally validate their properties. In essence, our approach boils down to constructing new families of linear codes with (plausibly) high minimum distance and extremely low encoding time. While further analysis is of course warranted to confidently assess the security of Silver, we hope and believe that initiating this approach to the design of MPC primitives will pave the way to new secure primitives with extremely attractive efficiency features.