Verifiable Secret Sharing (VSS) is a foundational cryptographic primitive that serves as an essential building block in multi-party computation and decentralized blockchain applications. One of the most practical ways to construct VSS is through a polynomial commitment, where the dealer commits to a random polynomial whose 0-th coefficient encodes the secret to be shared, and proves the evaluation of the committed polynomial at a different point to each of N verifiers, i.e., the polynomial commitment is used in a “one-to-many” fashion. The recent work of Tomescu et al. (IEEE S&P 2020) was the first to consider polynomial commitment with “one-to- many prover batching”, such that the prover can prove evaluations at N different points at the cost of O(1) proofs. However, their scheme is not optimal and requires a trusted setup.
In this paper, we asymptotically improve polynomial commitment with one-to-many prover batching. We propose two novel schemes. First, we propose a scheme with optimal asymptotics in all dimensions in the trusted setup setting. Second, we are the first to consider one-to-many prover batching for transparent polynomial commitments, and we propose a transparent scheme whose performance approximately matches the best-known scheme in the trusted setup setting. We implement our schemes and evaluate their performance. Our scheme in the trusted setup setting improves the proof size by 20× and the verifier time by 7.8× for 221 parties, with a small overhead on the prover time. Our transparent polynomial commitment removes the trusted setup and further improves the prover time by 2.3×.