Fast RS-IOP Multivariate Polynomial Commitments and Verifiable Secret Sharing

Supporting proofs of evaluations, polynomial commitment schemes (PCS) are crucial in secure distributed systems. Schemes based on fast Reed–Solomon interactive oracle proofs (RS-IOP) of proximity have recently emerged, offering transparent setup, plausible post-quantum security, efficient operations, and, notably, sublinear proof size and verification. Manifesting a new paradigm, PCS with one-to-many proof can enhance the performance of (asynchronous) verifiable secret sharing ((A)VSS), a cornerstone in distributed computing, for proving multiple evaluations to multiple verifiers. Current RS-IOP-based multivariate PCS, including HyperPlonk (Eurocrypt '23) and Virgo (S&P '20), however, only offer quasi-linear prover complexity in the polynomial size.

We propose PolyFRIM, a fast RS-IOP-based multivariate PCS with optimal linear prover complexity, 5-25× faster than prior arts while ensuring competent proof size and verification. Heeding the challenging absence of FFT circuits for multivariate evaluation, PolyFRIM surpasses Zhang et al.'s (Usenix Sec. '22) one-to-many univariate PCS, accelerating proving by 4-7× and verification by 2-4× with 25% shorter proof. Leveraging PolyFRIM, we propose an AVSS scheme FRISS with a better efficiency tradeoff than prior arts from multivariate PCS, including Bingo (Crypto '23) and Haven (FC '21).