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Non-interactive Zero-Knowledge from Non-interactive Batch Arguments

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Advances in Cryptology – CRYPTO 2023 (CRYPTO 2023)

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Abstract

Zero-knowledge and succinctness are two important properties that arise in the study of non-interactive arguments. Previously, Kitagawa et al. (TCC 2020) showed how to obtain a non-interactive zero-knowledge (NIZK) argument for \(\textsf{NP}\) from a succinct non-interactive argument (SNARG) for \(\textsf{NP}\). In particular, their work demonstrates how to leverage the succinctness property from an argument system and transform it into a zero-knowledge property.

   In this work, we study a similar question of leveraging succinctness for zero-knowledge. Our starting point is a batch argument for \(\textsf{NP}\), a primitive that allows a prover to convince a verifier of T \(\textsf{NP}\) statements \(x_1, \ldots , x_T\) with a proof whose size scales sublinearly with T. Unlike SNARGs for \(\textsf{NP}\), batch arguments for \(\textsf{NP}\) can be built from group-based assumptions in both pairing and pairing-free groups and from lattice-based assumptions. The challenge with batch arguments is that the proof size is only amortized over the number of instances, but can still encode full information about the witness to a small number of instances.

   We show how to combine a batch argument for \(\textsf{NP}\) with a local pseudorandom generator (i.e., a pseudorandom generator where each output bit only depends on a small number of input bits) and a dual-mode commitment scheme to obtain a NIZK for \(\textsf{NP}\). Our work provides a new generic approach of realizing zero-knowledge from succinctness and highlights a new connection between succinctness and zero-knowledge.

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Notes

  1. 1.

    More recently, Chakraborty et al.  [CPW23] showed a similar implication holds starting from a mildly-compact computational witness map (a simpler primitive that is implied by a SNARG for \(\textsf{NP}\)).

  2. 2.

    For instance, this is satisfied if the length of the proof scales polylogarithmically in the number of instances. More generally, we can instantiate the theorem even if the proof size scales with \(T^{1/2 - \varepsilon }\), where T is the number of instances and \(\varepsilon > 0\) is a constant. We refer to Corollary 3.15 for the precise characterization.

  3. 3.

    Semi-adaptive soundness for a batch argument says that the adversary must first commit to the index i of the false statement in the soundness game. It can adaptively choose the statements \(x_1, \ldots , x_T\) after seeing the CRS, with the restriction that instance \(x_i\) must be false.

  4. 4.

    In some settings, we can require a stronger “equivocation” property in hiding mode where given trapdoor information, one can sample a commitment c and openings for c to any value. Our constructions do not require equivocation.

  5. 5.

    Previous works [KPY19, CJJ21a, CJJ21b, WW22, DGKV22, CGJ+22] also impose requirements on the size of the CRS and the running time of the verifier. These additional properties are not needed in our work.

  6. 6.

    As noted above, we require that \(\ell (n) \ge m B\). If \(\ell (n) > m B\), we truncate the output of \(G_{n}\) to output a string of length exactly mB.

  7. 7.

    We note here that additionally assuming a rate-1 string oblivious transfer protocol [DGI+19], such a BARG can be transformed into a BARG where the proof size scales polylogarithmically with the number of instances [KLVW23]. In this case, we would be able to appeal to our previous instantiation.

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Acknowledgments

D. J. Wu is supported by NSF CNS-2151131, CNS-2140975, a Microsoft Research Faculty Fellowship, and a Google Research Scholar award.

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    Jeffrey Champion & David J. Wu

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Champion, J., Wu, D.J. (2023). Non-interactive Zero-Knowledge from Non-interactive Batch Arguments. In: Handschuh, H., Lysyanskaya, A. (eds) Advances in Cryptology – CRYPTO 2023. CRYPTO 2023. Lecture Notes in Computer Science, vol 14082. Springer, Cham. https://doi.org/10.1007/978-3-031-38545-2_2

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