Benedikt Bünz

Jianqiao Mo, Alhad Daftardar, Barath GaneshKumar et al.

In the ZKP community, it has long been discussed that the SumCheck protocol is asymptotically more efficient than the Number Theoretic Transform (NTT), requiring only $O(N)$ arithmetic versus $O(N \log N)$. At the same time, hardware accelerator designers propose that NTT is more hardware-friendly, benefiting from locality and data reuse, while SumCheck suffers from sequential, dependent rounds. Despite these competing intuitions, the hardware-system-level trade-offs between NTT- and SumCheck-based proving primitives remain insufficiently understood. Beyond individual accelerator design, this work presents, to our knowledge, the first hardware-system-level direct comparison of NTT- and SumCheck-based proving primitives under a unified architectural framework. We study them in the context of the ZeroCheck protocol, a common building block in zkSNARKs. We implement optimized systems for both primitives. Both are evaluated under the same level on-chip SRAM and off-chip bandwidth budgets. Our results show that there is no universal winner. Generally, SumCheck outperforms NTT for high-degree polynomials. For low-degree polynomials, performance depends on memory availability: under given SRAM budgets, NTT might deliver better performance for medium-sized workloads by exploiting data reuse. These findings, bridging cryptographic protocol design and hardware architecture, offer practical guidance for understanding the proving cost of NTT- and SumCheck-based zero-knowledge proof systems.

Daniel Selsam, Matthew Lamm, Benedikt Bünz et al.

We present NeuroSAT, a message passing neural network that learns to solve SAT problems after only being trained as a classifier to predict satisfiability. Although it is not competitive with state-of-the-art SAT solvers, NeuroSAT can solve problems that are substantially larger and more difficult than it ever saw during training by simply running for more iterations. Moreover, NeuroSAT generalizes to novel distributions; after training only on random SAT problems, at test time it can solve SAT problems encoding graph coloring, clique detection, dominating set, and vertex cover problems, all on a range of distributions over small random graphs.

Benedikt Bünz, Matthew Lamm

In this paper we explore whether or not deep neural architectures can learn to classify Boolean satisfiability (SAT). We devote considerable time to discussing the theoretical properties of SAT. Then, we define a graph representation for Boolean formulas in conjunctive normal form, and train neural classifiers over general graph structures called Graph Neural Networks, or GNNs, to recognize features of satisfiability. To the best of our knowledge this has never been tried before. Our preliminary findings are potentially profound. In a weakly-supervised setting, that is, without problem specific feature engineering, Graph Neural Networks can learn features of satisfiability.