Dual-Mode NIZKs from Obfuscation

Two standard security properties of a non-interactive zero-knowledge (NIZK) scheme are soundness and zero-knowledge. But while standard NIZK systems can only provide one of those properties against unbounded adversaries, dual-mode NIZK systems allow to choose dynamically and adaptively which of these properties holds unconditionally. The only known dual-mode NIZK systems are Groth-Sahai proofs (which have proved extremely useful in a variety of applications), and the FHE-based NIZK constructions of Canetti et al. and Peikert et al, which are concurrent and independent to this work. However, all these constructions rely on specific algebraic settings. Here, we provide a generic construction of dual-mode NIZK systems for all of NP. The public parameters of our scheme can be set up in one of two indistinguishable ways. One way provides unconditional soundness, while the other provides unconditional zero-knowledge. Our scheme relies on subexponentially secure indistinguishability obfuscation and subexponentially secure one-way functions, but otherwise only on comparatively mild and generic computational assumptions. These generic assumptions can be instantiated under any one of the DDH, k-LIN, DCR, or QR assumptions. As an application, we reduce the required assumptions necessary for several recent obfuscation-based constructions of multilinear maps. Combined with previous work, our scheme can be used to construct multilinear maps from obfuscation and a group in which the strong Diffie-Hellman assumption holds. We also believe that our work adds to the understanding of the construction of NIZK systems, as it provides a conceptually new way to achieve dual-mode properties

Non-Interactive Zero-Knowledge from Non-Interactive Batch Arguments

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 NP from a succinct non-interactive argument (SNARG) for 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 NP, a primitive that allows a prover to convince a verifier of $T$ NP statements $x_1, \ldots, x_T$ with a proof whose size scales sublinearly with $T$. Unlike SNARGs for NP, batch arguments for 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 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 NP. Our work provides a new generic approach of realizing zero-knowledge from succinctness and highlights a new connection between succinctness and zero-knowledge

Dual-Mode NIZKs: Possibility and Impossibility Results for Property Transfer

This paper formulates, and studies, the problem of property transference in dual-mode NIZKs. We say that a property P (such as soundness, ZK or WI) transfers, if, one of the modes having P allows us to prove that the other mode has the computational analogue of P, as a consequence of nothing but the indistinguishability of the CRSs in the two modes. Our most interesting finding is negative; we show by counter-example that the form of soundness that seems most important for applications fails to transfer. On the positive side, we develop a general framework that allows us to show that zero knowledge, witness indistinguishability, extractability and weaker forms of soundness do transfer. Our treatment covers conventional, designated-verifier and designated-prover NIZKs in a unified way

Correlation-Intractable Hash Functions via Shift-Hiding

A hash function family $\mathcal{H}$ is correlation intractable for a $t$-input relation $\mathcal{R}$ if, given a random function $h$ chosen from $\mathcal{H}$, it is hard to find $x_1,\ldots,x_t$ such that $\mathcal{R}(x_1,\ldots,x_t,h(x_1),\ldots,h(x_t))$ is true. Among other applications, such hash functions are a crucial tool for instantiating the Fiat-Shamir heuristic in the plain model, including the only known NIZK for NP based on the learning with errors (LWE) problem (Peikert and Shiehian, CRYPTO 2019). We give a conceptually simple and generic construction of single-input CI hash functions from shift-hiding shiftable functions (Peikert and Shiehian, PKC 2018) satisfying an additional one-wayness property. This results in a clean abstract framework for instantiating CI, and also shows that a previously existing function family (PKC 2018) was already CI under the LWE assumption. In addition, our framework transparently generalizes to other settings, yielding new results: - We show how to instantiate certain forms of multi-input CI under the LWE assumption. Prior constructions either relied on a very strong ``brute-force-is-best\u27\u27 type of hardness assumption (Holmgren and Lombardi, FOCS 2018) or were restricted to ``output-only\u27\u27 relations (Zhandry, CRYPTO 2016). - We construct single-input CI hash functions from indistinguishability obfuscation (iO) and one-way permutations. Prior constructions relied essentially on variants of fully homomorphic encryption that are impossible to construct from such primitives. This result also generalizes to more expressive variants of multi-input CI under iO and additional standard assumptions

Compact NIZKs from Standard Assumptions on Bilinear Maps

A non-interactive zero-knowledge (NIZK) protocol enables a prover to convince a verifier of the truth of a statement without leaking any other information by sending a single message. The main focus of this work is on exploring short pairing-based NIZKs for all NP languages based on standard assumptions. In this regime, the seminal work of Groth, Ostrovsky, and Sahai (J.ACM\u2712) (GOS-NIZK) is still considered to be the state-of-the-art. Although fairly efficient, one drawback of GOS-NIZK is that the proof size is multiplicative in the circuit size computing the NP relation. That is, the proof size grows by $O(|C|\lambda)$, where $C$ is the circuit for the NP relation and $\lambda$ is the security parameter. By now, there have been numerous follow-up works focusing on shortening the proof size of pairing-based NIZKs, however, thus far, all works come at the cost of relying either on a non-standard knowledge-type assumption or a non-static $q$-type assumption. Specifically, improving the proof size of the original GOS-NIZK under the same standard assumption has remained as an open problem. Our main result is a construction of a pairing-based NIZK for all of NP whose proof size is additive in $|C|$, that is, the proof size only grows by |C| +\poly(\lambda), based on the decisional linear (DLIN) assumption. Since the DLIN assumption is the same assumption underlying GOS-NIZK, our NIZK is a strict improvement on their proof size. As by-products of our main result, we also obtain the following two results: (1) We construct a perfectly zero-knowledge NIZK (NIPZK) for NP relations computable in NC1 with proof size |w| \cdot \poly(\lambda) where $|w|$ is the witness length based on the DLIN assumption. This is the first pairing-based NIPZK for a non-trivial class of NP languages whose proof size is independent of $|C|$ based on a standard assumption. (2)~We construct a universally composable (UC) NIZK for NP relations computable in NC1 in the erasure-free adaptive setting whose proof size is |w| \cdot \poly(\lambda) from the DLIN assumption. This is an improvement over the recent result of Katsumata, Nishimaki, Yamada, and Yamakawa (CRYPTO\u2719), which gave a similar result based on a non-static $q$-type assumption. The main building block for all of our NIZKs is a constrained signature scheme with decomposable online-offline efficiency. This is a property which we newly introduce in this paper and construct from the DLIN assumption. We believe this construction is of an independent interest

Security-Preserving Distributed Samplers: How to Generate any CRS in One Round without Random Oracles

A distributed sampler is a way for several mutually distrusting parties to non-interactively generate a common reference string (CRS) that all parties trust. Previous work constructs distributed samplers in the random oracle model, or in the standard model with very limited security guarantees. This is no accident, as standard model distributed samplers with full security were shown impossible. In this work, we provide new definitions for distributed samplers which we show achieve meaningful security guarantees in the standard model. In particular, our notion implies that the hardness of a wide range of security games is preserved when the CRS is replaced with a distributed sampler. We also show how to realize our notion of distributed samplers. A core technical tool enabling our construction is a new notion of single-message zero knowledge

New Constructions of Statistical NIZKs: Dual-Mode DV-NIZKs and More

International audienceNon-interactive zero-knowledge proofs (NIZKs) are important primitives in cryptography. A major challenge since the early works on NIZKs has been to construct NIZKs with a statistical zero-knowledge guarantee against unbounded verifiers. In the common reference string (CRS) model, such "statistical NIZK arguments" are currently known from k-Lin in a pairing-group and from LWE. In the (reusable) designated-verifier model (DV-NIZK), where a trusted setup algorithm generates a reusable verification key for checking proofs, we also have a construction from DCR. If we relax our requirements to computational zero-knowledge, we additionally have NIZKs from factoring and CDH in a pairing group in the CRS model, and from nearly all assumptions that imply public-key encryption (e.g., CDH, LPN, LWE) in the designated-verifier model. Thus, there still remains a gap in our understanding of statistical NIZKs in both the CRS and the designated-verifier models. In this work, we develop new techniques for constructing statistical NIZK arguments. First, we construct statistical DV-NIZK arguments from the k-Lin assumption in pairing-free groups, the QR assumption, and the DCR assumption. These are the first constructions in pairing-free groups and from QR that satisfy statistical zero-knowledge. All of our constructions are secure even if the verification key is chosen maliciously (i.e., they are "malicious-designated-verifier" NIZKs), and moreover, they satisfy a "dual-mode" property where the CRS can be sampled from two computationally indistinguishable distributions: one distribution yields statistical DV-NIZK arguments while the other yields computational DV-NIZK proofs. We then show how to adapt our k-Lin construction in a pairing group to obtain new publicly-verifiable statistical NIZK arguments from pairings with a qualitatively weaker assumption than existing constructions of pairing-based statistical NIZKs. Our constructions follow the classic paradigm of Feige, Lapidot, and Shamir (FLS). While the FLS framework has traditionally been used to construct computational (DV)-NIZK proofs, we newly show that the same framework can be leveraged to construct dual-mode (DV)-NIZKs

Multilinear Maps from Obfuscation

International audienceWe provide constructions of multilinear groups equipped with natural hard problems from in-distinguishability obfuscation, homomorphic encryption, and NIZKs. This complements known results on the constructions of indistinguishability obfuscators from multilinear maps in the reverse direction. We provide two distinct, but closely related constructions and show that multilinear analogues of the DDH assumption hold for them. Our first construction is symmetric and comes with a κ-linear map e : G κ −→ G T for prime-order groups G and G T. To establish the hardness of the κ-linear DDH problem, we rely on the existence of a base group for which the (κ − 1)-strong DDH assumption holds. Our second construction is for the asymmetric setting, where e : G 1 × · · · × G κ −→ G T for a collection of κ + 1 prime-order groups G i and G T , and relies only on the standard DDH assumption in its base group. In both constructions the linearity κ can be set to any arbitrary but a priori fixed polynomial value in the security parameter. We rely on a number of powerful tools in our constructions: (probabilistic) indistinguishability obfuscation, dual-mode NIZK proof systems (with perfect soundness, witness indistinguishability and zero knowledge), and additively homomorphic encryption for the group Z + N. At a high level, we enable " bootstrapping " multilinear assumptions from their simpler counterparts in standard cryptographic groups, and show the equivalence of IO and multilinear maps under the existence of the aforementioned primitives

Non-Interactive Proofs: What Assumptions Are Sufficient?

A non-Interactive proof system allows a prover to convince a verifier that a statement is true by sending a single round of messages. In this thesis, we study under what assumptions can we build non-interactive proof systems with succinct verification and zero-knowledge. We obtain the following results. - Succinct Arguments: We construct the first non-interactive succinct arguments (SNARGs) for P from standard assumptions. Our construction is based on the polynomial hardness of Learning with Errors (LWE). - Zero-Knowledge: We build the first non-interactive zero-knowledge proof systems (NIZKs) for NP from sub-exponential Decisional Diffie-Hellman (DDH) assumption in the standard groups, without use of groups with pairings. To obtain our results, we build SNARGs for batch-NP from LWE and correlation intractable hash functions for TC^0 from sub-exponential DDH assumption, respectively, which may be of independent interest

Classically Verifiable NIZK for QMA with Preprocessing

We propose three constructions of classically verifiable non-interactive zero-knowledge proofs and arguments (CV-NIZK) for QMA in various preprocessing models. 1. We construct a CV-NIZK for QMA in the quantum secret parameter model where a trusted setup sends a quantum proving key to the prover and a classical verification key to the verifier. It is information theoretically sound and zero-knowledge. 2. Assuming the quantum hardness of the learning with errors problem, we construct a CV-NIZK for QMA in a model where a trusted party generates a CRS and the verifier sends an instance-independent quantum message to the prover as preprocessing. This model is the same as one considered in the recent work by Coladangelo, Vidick, and Zhang (CRYPTO \u2720). Our construction has the so-called dual-mode property, which means that there are two computationally indistinguishable modes of generating CRS, and we have information theoretical soundness in one mode and information theoretical zero-knowledge property in the other. This answers an open problem left by Coladangelo et al, which is to achieve either of soundness or zero-knowledge information theoretically. To the best of our knowledge, ours is the first dual-mode NIZK for QMA in any kind of model. 3. We construct a CV-NIZK for QMA with quantum preprocessing in the quantum random oracle model. This quantum preprocessing is the one where the verifier sends a random Pauli-basis states to the prover. Our construction uses the Fiat-Shamir transformation. The quantum preprocessing can be replaced with the setup that distributes Bell pairs among the prover and the verifier, and therefore we solve the open problem by Broadbent and Grilo (FOCS \u2720) about the possibility of NIZK for QMA in the shared Bell pair model via the Fiat-Shamir transformation