We propose the first collaborative SNARK based on error-correcting codes that is scalable, i.e., the proof computation overhead is distributed among the N provers. As a starting point, we introduce the notion of (t, l)-zero-knowledge collaborative codes that ensure that, when collaboratively computing a codeword over a distributed message, no coalition of up to t corrupted parties learns any additional information about the message, even having queried up to l codeword positions. We show that tensor codes consisting of the composition of two Reed-Solomon codes satisfy our definition, while also being foldable. We then propose a collaborative interactive oracle proof of proximity (coIOPP) for testing codeword closeness in our code, show how it can be made a zero-knowledge IOPP using randomness logarithmic in the size of the message (as opposed to linear with prior approaches), and we use it to construct a coIOPP for multi-linear polynomial evaluation. To compile our coIOPPs into non-interactive arguments, we prove that a natural extension of the compiler of Ben-Sasson-ChiesaSpooner (TCC 2016) in the collaborative setting preserves round-by-round (knowledge) soundness against quantum adversaries, which may be of independent interest for future work in collaborative SNARKs. Finally, we use an optimized collaborative version of the Spartan PIOP to build the first transparent and post-quantum secure scalable collaborative SNARK. Our experimental evaluation demonstrates that our scheme consistently outperforms the best existing (non-postquantum secure) scalable collaborative SNARKs, both in end-to-end prover time and in total communication among provers, for all tested configurations.

We address the challenge of constructing a proof system capable of handling multiple computations that involve diverse types of tasks, such as scalable zkRollup applications. A central dilemma in this design is the trade-off between generality and efficiency: while arithmetic circuit-based SNARKs offer fast proofs but limited flexibility, zkVMs provide general-purpose programmability at the cost of considerable overhead for circuit translation. We observe that typical workloads for such applications can be naturally divided into two parts: (1) diverse, task and data-dependent application logic, and (2) computationally intensive cryptographic operations, e.g., hashes, that are common and repetitive. To optimize for both efficiency and adaptability, we propose Gryphes, a hybrid framework that composes matrix lookup, a generalization of lookup arguments, together with SNARK solutions tailored for cryptographic operations. At the heart of ame is a novel and efficient linking protocol, enabling seamless, efficient composition of matrix lookup + Plonk with general commit-and-prove SNARKs. By integrating Gryphes with Groth16 for signatures and RSA accumulators for membership proofs, we build a zkRollup prototype that achieves efficient proving, constant-size proofs, and dynamic support for thousands of transaction types. This includes our matrix lookup implementation incorporated with Plonk, as well as practical optimizations, comprehensive benchmarks, and open-sourced code. Our results demonstrate that Gryphes strikes a very good balance between functionality and efficiency, offering highly expressive and practical zkRollup systems.

In this work, we introduce HydraProofs, the first vector commitment (VC) scheme that achieves the following two properties. (i) The prover can produce all the opening proofs for different elements (or consecutive sub-arrays) for a vector of size$N$in optimal time$\mathcal{O}(N)$. (ii) It is directly compatible with a family of zkSNARKs that encode their input as a multi-linear polynomial, i.e., our VC can be directly used when running the zkSNARK on its pre-image, without the need to “open” the entire vector pre-image inside the zkSNARK. To the best of our knowledge, all prior VC schemes either achieve (i) but are not efficiently “pluggable” into zkSNARKs (e.g., a Merkle tree commitment that requires re-computing the entire hash tree inside the circuit), or achieve (ii) but take$\mathrm{O}(N\log N)$time. We then combine HydraProofs with the seminal GKR protocol and apply the resulting zkSNARK in a setting where multiple users participate in a computation executed by an untrusted server and each user wants to ensure the correctness of the result and that her data was included. Our experimental evaluation shows our approach outperforms prior ones by 4 - 16× for prover times on general circuits. Finally, we consider two concrete application use cases, verifiable secret sharing and verifiable robust aggregation. For the former, our construction achieves the first scheme for Shamir's secret sharing with linear time prover (lower than the time needed for the dealer computation). For the second, we propose a scheme that works against misbehaving aggregators and our experiments show it can be reasonably deployed in existing schemes with minimal slow-downs.

A zero-knowledge proof of training (zkPoT) enables a party to prove that they have correctly trained a committed model based on a committed dataset without revealing any additional information about the model or the dataset. An ideal zkPoT should offer provable security and privacy guarantees, succinct proof size and verifier runtime, and practical prover efficiency. In this work, we present Kaizen, a zkPoT targeted for deep neural networks (DNNs) that achieves all these goals at once. Our construction enables a prover to iteratively train their model via (mini-batch) gradient descent, where the number of iterations need not be fixed in advance; at the end of each iteration, the prover generates a commitment to the trained model parameters attached with a succinct zkPoT, attesting to the correctness of the executed iterations. The proof size and verifier time are independent of the number of iterations.

Space-efficient SNARKs aim to reduce the prover's space overhead which is one the main obstacles for deploying SNARKs in practice, as it can be prohibitively large (e.g., orders of magnitude larger than natively performing the computation). In this work, we propose Sparrow, a novel space-efficient zero-knowledge SNARK for data-parallel arithmetic circuits with two attractive features: (i) it is the first space-efficient scheme where, for a given field, the prover overhead increases with a multiplicative sublogarithmic factor as the circuit size increases, and (ii) compared to prior space-efficient SNARKs that work for arbitrary arithmetic circuits, it achieves prover space asymptotically smaller than the circuit size itself. Our key building block is a novel space-efficient sumcheck argument with improved prover time which may be of independent interest. Our experimental results for three use cases (arbitrary data parallel circuits, multiplication trees, batch SHA256 hashing) indicate Sparrow outperforms the prior state-of-the-art space-efficient SNARK for arithmetic circuits Gemini (Bootle et al., EUROCRYPT'22) by 3.2-28.7x in total prover space and 3.1-11.3x in prover time. We then use Sparrow to build zero-knowledge proofs of tree training and prediction, relying on its space efficiency to scale to large datasets and forests of multiple trees. Compared to a (non-space-efficient) optimal-time SNARK based on the GKR protocol, we observe prover space reduction of 16-240x for tree training while maintaining essentially the same prover and verifier times and proof size. Even more interestingly, our prover requires comparable space to natively perform the underlying computation. E.g., for a 400MB dataset, our prover only needs 1.4x more space than the native computation.

The proliferation of resourceful mobile devices that store rich, multidimensional and privacy-sensitive user data motivate federated learning, a paradigm that enables mobile devices to produce a machine-learning model without sharing their data. However, the majority of the existing federated frameworks follow a centralized approach. In this work, we introduce IPLS, a fully decentralized federated learning framework that is partially based on the interplanetary file system (IPFS). By using IPLS and connecting into the corresponding private IPFS network, any party can initiate the training process of a machine-learning model or join an ongoing training process that has been started by another party. IPLS scales with the number of participants, is robust against intermittent connectivity and dynamic participant departures/arrivals, requires minimal resources and guarantees that the accuracy of the trained model quickly converges to that of a centralized federated learning framework with a negligible accuracy drop of less than 10/00.