International Association
for Cryptologic Research

We construct the first polynomial commitment scheme (PCS) that has a transparent setup, quasi-linear prover time, $\log N$ verifier time, and $\log \log N$ proof size, for multilinear polynomials of size $N$. Concretely, we have the smallest proof size amongst transparent PCS, with proof size less than $4.5$KB for $N\leq 2^{30}$. We prove that our scheme is secure entirely under falsifiable assumptions about groups of unknown order. The scheme significantly improves on the prior work of Dew (PKC 2023), which has super-cubic prover time and relies on the Generic Group Model (a non-falsifiable assumption). Along the way, we make several contributions that are of independent interest: $\PoKEMath$, a protocol for efficiently proving that an arbitrary predicate over committed integer vectors holds; $\SIPA$, a bulletproofs-style inner product argument in groups of unknown order; we also distill out what prior work required from the Generic Group Model and frame this as a falsifiable assumption.

Modern SNARK designs typically follow a frontend-backend paradigm: The frontend compiles a user's program into some equivalent circuit representation, while the backend calls for a SNARK specifically made for proving circuit satisfiability. While these circuits are often defined over small fields, the backend prover always needs to lift the computation to much larger fields to ensure soundness. This gap introduces concrete overheads for ZK applications like zkRollups, where group-based SNARKs are used to provide constant-size proofs for Merkle tree openings. For a class of highly repetitive computations, we propose FREPack, an improved frontend that effectively bridges this gap. The larger the gap between circuit's small field and backend's large field, the more FREPack reduces the circuit size, making it particularly well-suited for group-based backends. Our implementation shows that, for proving ~ 300 iterations of SHA-256, FREPack improves the performance of Groth16 by 3.6x, Nova by 3.8x, and Spartan by 5.9x.

We construct a polynomial commitment scheme with constant (i.e., independent of the degree) sized evaluation proofs and logarithmic (in the degree) verification time in the transparent setting. To the best of our knowledge, this is the first result achieving this combination of properties. We build our scheme from an inner product commitment scheme with constant-sized proofs but with linear verification time. To improve the verification time to logarithmic for polynomial commitments, we prove a new extremal combinatorial bound. Our constructions rely on groups of unknown order instantiated by class groups. We prove security of our constructions in the Generic Group Model. Compiling known information-theoretic proof systems using our polynomial commitment scheme yields transparent and constant-sized zkSNARKs (Zero-knowledge Succinct Non-interactive ARguments of Knowledge) with logarithmic verification.