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2018 | OriginalPaper | Buchkapitel

Multi-Theorem Preprocessing NIZKs from Lattices

verfasst von : Sam Kim, David J. Wu

Erschienen in: Advances in Cryptology – CRYPTO 2018

Verlag: Springer International Publishing

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Abstract

Non-interactive zero-knowledge (NIZK) proofs are fundamental to modern cryptography. Numerous NIZK constructions are known in both the random oracle and the common reference string (CRS) models. In the CRS model, there exist constructions from several classes of cryptographic assumptions such as trapdoor permutations, pairings, and indistinguishability obfuscation. Notably absent from this list, however, are constructions from standard lattice assumptions. While there has been partial progress in realizing NIZKs from lattices for specific languages, constructing NIZK proofs (and arguments) for all of \(\mathsf {NP}\) from standard lattice assumptions remains open.

In this work, we make progress on this problem by giving the first construction of a multi-theorem NIZK argument for \(\mathsf {NP}\) from standard lattice assumptions in the preprocessing model. In the preprocessing model, a (trusted) setup algorithm generates proving and verification keys. The proving key is needed to construct proofs and the verification key is needed to check proofs. In the multi-theorem setting, the proving and verification keys should be reusable for an unbounded number of theorems without compromising soundness or zero-knowledge. Existing constructions of NIZKs in the preprocessing model (or even the designated-verifier model) that rely on weaker assumptions like one-way functions or oblivious transfer are only secure in a single-theorem setting. Thus, constructing multi-theorem NIZKs in the preprocessing model does not seem to be inherently easier than constructing them in the CRS model.

We begin by constructing a multi-theorem preprocessing NIZK directly from context-hiding homomorphic signatures. Then, we show how to efficiently implement the preprocessing step using a new cryptographic primitive called blind homomorphic signatures. This primitive may be of independent interest. Finally, we show how to leverage our new lattice-based preprocessing NIZKs to obtain new malicious-secure MPC protocols purely from standard lattice assumptions.

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Vorheriges Kapitel Lattice-Based Zero-Knowledge Arguments for Integer Relations

Nächstes Kapitel SPD: Efficient MPC mod for Dishonest Majority

There are also NIZK candidates based on number-theoretic assumptions [15, 16, 41] which satisfy weaker properties. We discuss these in greater detail in Sect. 1.2 and Remark 4.7.

This is a classic technique in the construction of non-interactive proof systems and has featured in many contexts (e.g., [56, 87]).

Since it is more natural to view \(x \in \{0,1\}^n\) as a string rather than a vector, we drop the vector notation \(\varvec{x}\) and simply write x in this section.

Some of these schemes [16, 41] are “bounded” in the sense that the prover can only prove a small number of theorems whose total size is bounded by the length of the CRS.

At a high-level, the proof in [44] proceeds in two steps: first show that single-theorem zero knowledge implies single-theorem witness indistinguishability, and then that single-theorem witness indistinguishability implies multi-theorem witness indistinguishability. The second step relies on a hybrid argument, which requires that it be possible to publicly run the prover algorithm. This step does not go through if the prover algorithm takes in a secret state unknown to the verifier.

If there is no binding between \(\sigma ^*\) and the function g, then we cannot define a meaningful notion of unforgeability.

The adversary is not allowed to re-register a signature that was previously declared invalid (according to the verification functionality) as a valid signature.

For the protocol description and its security proof, we use the vector notation \(\varvec{x}\) to represent the messages (in order to be consistent with the homomorphic signature notation).

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Titel: Multi-Theorem Preprocessing NIZKs from Lattices
verfasst von: Sam Kim
David J. Wu
Verlag: Springer International Publishing
Buch: Advances in Cryptology – CRYPTO 2018
Print ISBN: 978-3-319-96880-3

Electronic ISBN: 978-3-319-96881-0

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-96881-0_25

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