DeepFold: Efficient Multilinear Polynomial Commitment from Reed-Solomon Code and Its Application to Zero-knowledge Proofs

Yanpei Guo, Xuanming Liu, Kexi Huang, Wenjie Qu, Tianyang Tao, Jiaheng Zhang

Abstract: This work presents Deepfold, a novel multilinear polynomial commitment scheme (PCS) based on Reed-Solomon code that offers optimal prover time and a more concise proof size. For the first time, Deepfold adapts the FRI-based multilinear PCS to the list decoding radius setting, requiring significantly fewer query repetitions and thereby achieving a 3$\times$ reduction in proof size compared to Basefold (Crypto'24), while preserving its advantages in prover time. Compared with PolyFRIM (USENIX Security'24), Deepfold achieves a 2$\times$ improvement in prover time, verifier time, and proof size. Another contribution of this work is a batch evaluation scheme, which enables the FRI-based multilinear PCS to handle polynomials whose size is not a power of two more efficiently. Our scheme has broad applications in zk-SNARKs, since PCS is a key component in modern zk-SNARK constructions. For example, when replacing the PCS component of Virgo (S&P'20) with Deepfold, our scheme achieves a 2.5$\times$ faster prover time when proving the knowledge of a Merkle tree with 256 leaves, while maintaining the similar proof size. When replacing the PCS component of HyperPlonk (Eurocrypt'23) with Deepfold, our scheme has about 3.6$\times$ faster prover time. Additionally, when applying our arbitrary length input commitment to verifiable matrix multiplications for matrices of size 1200$\times$768 and 768$\times$2304, which are actual use cases in GPT-2 model, the performance showcases a 2.4$\times$ reduction in prover time compared to previous approaches.