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Efficient Zero-Knowledge Arguments in Discrete Logarithm Setting: Sublogarithmic Proof or Sublinear Verifier

Part of the Lecture Notes in Computer Science book series (LNCS,volume 13792)

Abstract

We propose three interactive zero-knowledge arguments for arithmetic circuit of size N in the common random string model, which can be converted to be non-interactive by Fiat-Shamir heuristics in the random oracle model. First argument features \(O(\sqrt{\log N})\) communication and round complexities and O(N) computational complexity for the verifier. Second argument features \(O(\log N)\) communication and \(O(\sqrt{N})\) computational complexity for the verifier. Third argument features \(O(\log N)\) communication and \(O(\sqrt{N}\log N)\) computational complexity for the verifier. Contrary to first and second arguments, the third argument is free of reliance on pairing-friendly elliptic curves. The soundness of three arguments is proven under the standard discrete logarithm and/or the double pairing assumption, which is at least as reliable as the decisional Diffie-Hellman assumption.

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Notes

  1. 1.

    We often use a terminology ‘transparent’ in the meaning of ‘without trusted setup’.

  2. 2.

    Note that we use the indices \((1,-1)\) instead of (1, 2) since it harmonizes well with the usage of the challenges in Bulletproofs and our generalization of Bulletproofs. e.g., let \({\boldsymbol{a}}={\boldsymbol{a}}_1\Vert {\boldsymbol{a}}_{-1}\) be a witness and x be a challenge, and then \({\boldsymbol{a}}\) is updated to \(\sum _{i=\pm 1}{\boldsymbol{a}}_i x^i\), a witness for the next recursive round.

  3. 3.

    The BP-IP is about two witness vectors \({\boldsymbol{a}}\) and \({\boldsymbol{b}}\) and it can be easily modified with one witness vector \({\boldsymbol{a}}\) and a public \({\boldsymbol{b}}\). e.g., [46]. Our multi-exponentiation argument corresponds to this variant.

  4. 4.

    Note that when the communication complexity is evaluated, we set \(n=2^{\sqrt{\log N}}\) that is much smaller than \(\sqrt{N}=2^{\frac{1}{2}\log N}\), and thus our estimation for computational cost makes sense.

  5. 5.

    Note that this operation is also called “projecting bilinear map" in the context of converting composite-order bilinear groups to prime-order bilinear groups [26].

  6. 6.

    Note that the DL assumption on \({\mathbb {G}}_1\) implies the DL assumption on \({\mathbb {G}}_t\) by the MOV attack [39].

  7. 7.

    If needed, we can appropriately pad zeros in the vectors since zeros do not affect the result of inner-product.

  8. 8.

    More precisely, we use a slightly modified Pedersen commitment scheme in the sense that (1) opening is not an integer but a vector and (2) the random element is always set to be zero since the hiding property is not required.

  9. 9.

    Notice that we use a subscript k in reverse order from \(k=\ell \) to \(k=1\). That is, \(\mathsf {Protocol4.Row}\) reduces an instance from \(P_{k+1}\) to \(P_{k}\).

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Acknowledgement

We thank Taechan Kim for discussion on complete addition formulas for elliptic curves. This work was supported in part by the Institute of Information and Communications Technology Planning and Evaluation (IITP) grant funded by the Korea Government (MSIT) (A Study on Cryptographic Primitives for SNARK, 50%) under Grant 20210007270012002, and in part by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIT), 50%, under Grant 2020R1C1C1A0100696812.

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Kim, S., Lee, H., Seo, J.H. (2022). Efficient Zero-Knowledge Arguments in Discrete Logarithm Setting: Sublogarithmic Proof or Sublinear Verifier. In: Agrawal, S., Lin, D. (eds) Advances in Cryptology – ASIACRYPT 2022. ASIACRYPT 2022. Lecture Notes in Computer Science, vol 13792. Springer, Cham. https://doi.org/10.1007/978-3-031-22966-4_14

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