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Lattice-Based Succinct Arguments for NP with Polylogarithmic-Time Verification

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Advances in Cryptology – CRYPTO 2023 (CRYPTO 2023)

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Abstract

Succinct arguments that rely on the Merkle-tree paradigm introduced by Kilian (STOC 92) suffer from larger proof sizes in practice due to the use of generic cryptographic primitives. In contrast, succinct arguments with the smallest proof sizes in practice exploit homomorphic commitments. However these latter are quantum insecure, unlike succinct arguments based on the Merkle-tree paradigm.

A recent line of works seeks to address this limitation, by constructing quantum-safe succinct arguments that exploit lattice-based commitments. The eventual goal is smaller proof sizes than those achieved via the Merkle-tree paradigm. Alas, known constructions lack succinct verification.

In this paper, we construct the first interactive argument system for NP with succinct verification that, departing from the Merkle-tree paradigm, exploits the homomorphic properties of lattice-based commitments. For an arithmetic circuit with N gates, our construction achieves verification time \(\textsf{polylog}(N)\) based on the hardness of the Ring Short-Integer-Solution (RSIS) problem.

The core technique in our construction is a delegation protocol built from commitment schemes based on leveled bilinear modules, a new notion that we deem of independent interest. We show that leveled bilinear modules can be realized from pre-quantum and from post-quantum cryptographic assumptions.

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Notes

  1. 1.

    Of course, in our lattice instantiation, \(M_{\scriptscriptstyle \textrm{R},i}\) and \(M_{\scriptscriptstyle \textrm{T},i}\) happen to be the same.

  2. 2.

    In more detail, consider a cyclotomic ring of the form \(R:=\mathbb {Z}[X]/\langle \varPhi _{d}(X)\rangle \) where \(\varPhi _{d}(X)\) is the \(d\)-th cyclotomic polynomial. The polynomial \(\varPhi _{d}(X)\) modulo a prime p with \(\gcd (p, d) = 1\) factors into irreducible polynomials of the same degree \(t\) for some \(t\in \mathbb {N}\) (e.g., from [31, Theorem 5.3]). This means that \(R/ pR\) is isomorphic to \(k:=\phi (d) / t\) copies of \(\mathbb {F}_{p^t}\).

  3. 3.

    This is despite the fact that the PIOP construction in full generality sometimes uses non-algebraic operations such as linear scans.

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Authors and Affiliations

  1. IBM Research Europe, Zurich, Switzerland

    Jonathan Bootle

  2. EPFL, Lausanne, Switzerland

    Alessandro Chiesa

  3. UC Berkeley, Berkeley, USA

    Katerina Sotiraki

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Bootle, J., Chiesa, A., Sotiraki, K. (2023). Lattice-Based Succinct Arguments for NP with Polylogarithmic-Time Verification. 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_8

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