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Zero-Knowledge IOPs with Linear-Time Prover and Polylogarithmic-Time Verifier

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Advances in Cryptology – EUROCRYPT 2022 (EUROCRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13276))

Abstract

Interactive oracle proofs (IOPs) are a multi-round generalization of probabilistically checkable proofs that play a fundamental role in the construction of efficient cryptographic proofs.

We present an IOP that simultaneously achieves the properties of zero knowledge, linear-time proving, and polylogarithmic-time verification. We construct a zero-knowledge IOP where, for the satisfiability of an N-gate arithmetic circuit over any field of size \(\varOmega (N)\), the prover uses O(N) field operations and the verifier uses \({\mathsf {polylog}}(N)\) field operations (with proof length O(N) and query complexity \({\mathsf {polylog}}(N)\)). Polylogarithmic verification is achieved in the holographic setting for every circuit (the verifier has oracle access to a linear-time-computable encoding of the circuit whose satisfiability is being proved).

Our result implies progress on a basic goal in the area of efficient zero knowledge. Via a known transformation, we obtain a zero knowledge argument system where the prover runs in linear time and the verifier runs in polylogarithmic time; the construction is plausibly post-quantum and only makes a black-box use of lightweight cryptography (collision-resistant hash functions).

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Notes

  1. 1.

    Several of these works additionally achieve excellent concrete efficiency, via experiments that demonstrate the ability to prove the satisfiability of circuits with billions of gates.

  2. 2.

    As soundness is computational then we can hope for zero knowledge to be statistical.

  3. 3.

    Satisfiability of an \(n\)-gate arithmetic circuit over the field \(\mathbb {F}\) is reducible, in linear time, to an R1CS instance also over \(\mathbb {F}\) where the coefficient matrices are \(n\times n\) and have \(m=O(n)\) non-zero entries. (In particular, the coefficient matrices are sparse.).

  4. 4.

    Note that \(m= \varOmega (n)\) without loss of generality because if \(m< n/3\) then there are variables of \(z\) that do not participate in any constraint, which can be dropped. Thus the main size measure for R1CS is the sparsity parameter \(m\).

  5. 5.

    The private coins come from using the Goldreich–Kahan technique [26]. Achieving public coins is also possible via different relaxations: (i) (ii) we could rely on a reference string (which enables the zero knowledge simulator to access a trapdoor); or (iii) we could relax the goal to honest-verifier zero-knowledge while remaining in the plain model. See [34] for more on these considerations.

  6. 6.

    Holography/preprocessing may be avoidable by focusing on R1CS instances with a short description [6] or, more generally, uniform models of computation. Achieving results analogous to ours in such a setting remains an open problem.

  7. 7.

    The quasilinear costs in some works (due to cryptography [53, 54] or an FFT [55]) scale with witness size rather than computation size, and so the prover runs in linear time when the witness is small relative to the computation.

  8. 8.

    Some of the cited works still refer to such prover time as “linear” or “asymptotically optimal”. This is a misnomer.

  9. 9.

    A proof system is robust if the local view of the verifier is far (e.g. in Hamming distance) from an accepting view with high probability (over the verifier’s randomness) whenever the instance is not in the language.

  10. 10.

    A proximity proof shows that a given input is close to some input in the language.

  11. 11.

    This is related to special honest-verifier zero-knowledge for sigma protocols.

  12. 12.

    Note also that query bound \(b\) must be at least the number of queries that \(\hat{\mathbf {V}}\) makes to the encoded proof \(\hat{\varPi }\).

  13. 13.

    An IOP is said to have robustness parameter \(\alpha \) if the local view of the verifier is \(\alpha \)-close (in relative Hamming distance) to an accepting view with probability bounded by the IOP’s soundness error.

  14. 14.

    Note that as described in [14], the tensor IOP of [14] achieves verifier complexity \(O(|{x}|+k)\) because some of the verifier’s tensor queries are generated from seeds of length \(O(k)\). We reduce the verifier complexity by generating the verifier’s tensor queries using short seeds.

  15. 15.

    For example, constructing linear PCPs that are zero knowledge against malicious verifiers remains an open problem. Constructing tensor IOPs that are zero knowledge against malicious verifiers, while formally an easier question, appears similarly hard.

  16. 16.

    This is distinct from how zero knowledge is achieved for prior IOPs for R1CS based on the Reed–Solomon code [7]. Instead, it is closer in spirit to how semi-honest-verifier zero knowledge was achieved for linear PCPs for circuits or quadratic arithmetic programs in [11, 23].

  17. 17.

    We stress that this modification achieves zero knowledge only against semi-honest verifiers, because a malicious verifier could choose to query the padded vectors with a linear combination that leaves out the randomness and thereby learns information about the secret witness. Nevertheless, as discussed in Sect. 2.3, a tensor IOP that is merely semi-honest-verifier zero knowledge suffices for obtaining a point-query IOP with zero knowledge against bounded-query malicious verifiers.

  18. 18.

    In Kilian’s approach, the argument prover’s cryptographic cost is dominated by the cost to commit to the PCP string via a Merkle tree. In particular, if the PCP has proof length \(\mathsf {l}\) and the size of a proof symbol is linear in the input size of the hash function, then the running time of the argument prover is within a constant of the running time of the PCP prover.

  19. 19.

    Modify the Merkle tree to be over hiding commitments to proof symbols (rather than over the proof symbols themselves) and then prove in zero knowledge that opening the queried locations would have made the probabilistic proof verifier accept.

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Authors and Affiliations

  1. IBM Research, Zurich, Switzerland

    Jonathan Bootle

  2. École polytechnique fédérale de Lausanne, Lausanne, Switzerland

    Alessandro Chiesa

  3. University of California, Berkeley, Berkeley, USA

    Alessandro Chiesa & Siqi Liu

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  1. Jonathan Bootle
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  2. Alessandro Chiesa
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Correspondence to Jonathan Bootle or Alessandro Chiesa .

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  1. University of Haifa, Haifa, Haifa, Israel

    Orr Dunkelman

  2. University of Warsaw, Warsaw, Poland

    Stefan Dziembowski

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Bootle, J., Chiesa, A., Liu, S. (2022). Zero-Knowledge IOPs with Linear-Time Prover and Polylogarithmic-Time Verifier. In: Dunkelman, O., Dziembowski, S. (eds) Advances in Cryptology – EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol 13276. Springer, Cham. https://doi.org/10.1007/978-3-031-07085-3_10

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