International Association for Cryptologic Research

International Association
for Cryptologic Research


Noah Vesely


Proofs for Inner Pairing Products and Applications 📺
We present a generalized inner product argument and demonstrate its applications to pairing-based languages. We apply our generalized argument to prove that an inner pairing product is correctly evaluated with respect to committed vectors of $n$ source group elements. With a structured reference string (SRS), we achieve a logarithmic-time verifier whose work is dominated by $6 \log n$ target group exponentiations. Proofs are of size $6 \log n$ target group elements, computed using $6n$ pairings and $4n$ exponentiations in each source group. We apply our inner product arguments to build the first polynomial commitment scheme with succinct (logarithmic) verification, $O(\sqrt{d})$ prover complexity for degree $d$ polynomials (not including the cost to evaluate the polynomial), and a SRS of size $O(\sqrt{d})$. Concretely, this means that for $d=2^{28}$, producing an evaluation proof in our protocol is $76\times$ faster than doing so in the KZG commitment scheme, and the CRS in our protocol is $1000\times$ smaller: $13$MB vs $13$GB for KZG. As a second application, we introduce an argument for aggregating $n$ Groth16 zkSNARKs into an $O(\log n)$ sized proof. Our protocol is significantly faster ($>1000\times$) than aggregating SNARKs via recursive composition: we aggregate $\sim 130,000$ proofs in $25$ minutes, versus $90$ proofs via recursive composition. Finally, we further apply our aggregation protocol to construct a low-memory SNARK for machine computations that does not rely on recursive composition. For a computation that requires time $T$ and space $S$, our SNARK produces proofs in space $\tilde{\mathcal{O}}(S+T)$, which is significantly more space efficient than a monolithic SNARK, which requires space $\tilde{\mathcal{O}}(S \cdot T)$.
Marlin: Preprocessing zkSNARKs with Universal and Updatable SRS 📺
We present a general methodology to construct preprocessing zkSNARKs where the structured reference string (SRS) is universal and updatable. This exploits a novel application of *holographic* IOPs, a natural generalization of holographic PCPs [Babai et al., STOC 1991]. We use our methodology to obtain a preprocessing zkSNARK where the SRS has linear size and arguments have constant size. Our construction improves on Sonic [Maller et al., CCS 2019], the prior state of the art in this setting, in all efficiency parameters: proving is an order of magnitude faster and verification is twice as fast, even with smaller SRS size and argument size. Our construction is most efficient when instantiated in the algebraic group model (also used by Sonic), but we also demonstrate how to realize it under concrete knowledge assumptions. The core of our zkSNARK is a new holographic IOP for rank-1 constraint satisfiability (R1CS), which is the first to achieve linear proof length and constant query complexity (among other efficiency features).