Paper 2019/1177

Proofs for Inner Pairing Products and Applications

Benedikt Bünz, Mary Maller, Pratyush Mishra, Nirvan Tyagi, and Psi Vesely


We present a generalized inner product argument and demonstrate its applications to pairing-based languages. We apply our generalized argument to proving 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 CRS 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 $1,000\times$ smaller: $13$MB vs $13$GB for KZG. This gap only grows as the degree increases. Our polynomial commitment scheme is applicable to both univariate and bivariate polynomials. As a second application, we introduce an argument for aggregating $n$ $\mathsf{Groth16}$ zkSNARKs into an $O(\log n)$ sized proof. Our protocol is significantly more efficient than aggregating these SNARKs via recursive composition (BCGMMW20): we can aggregate about $130,000$ proofs in $25$min, while in the same time recursive composition aggregates just $90$ proofs. Finally, we show how to apply our aggregation protocol to construct a low-memory SNARK for machine computations. 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 $\tilde{\mathcal{O}}(S \cdot T)$.

Note: Major update. Better presentation, full implementation and evaluation

Available format(s)
Public-key cryptography
Publication info
Preprint. MINOR revision.
inner pairing productbilinear pairingsaggregate signaturesaggregate proofsstructure-preserving cryptography
Contact author(s)
psi @ berkeley edu
mary maller @ ethereum org
benedikt @ cs stanford edu
pratyush @ berkeley edu
nirvan tyagi @ gmail com
2020-12-09: last of 9 revisions
2019-10-10: received
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Creative Commons Attribution


      author = {Benedikt Bünz and Mary Maller and Pratyush Mishra and Nirvan Tyagi and Psi Vesely},
      title = {Proofs for Inner Pairing Products and Applications},
      howpublished = {Cryptology ePrint Archive, Paper 2019/1177},
      year = {2019},
      note = {\url{}},
      url = {}
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