An Unbounded Simulation-Sound Non-interactive Zero-Knowledge Proof System for NP

Direct Anonymous Attestation (DAA) is a special signature scheme that enables remote authentication of a user while preserving privacy under the user's control. Thus, in this paper, based on existing schemes and mechanisms, we firstly construct a DAA signature scheme which is proven secure without random oracles. By making certain reasonable assumptions and employing the techniques of non-interactive zero-knowledge(NIZK) proof system and Two-Party Computation(2PC) scheme, we prove that our scheme satisfies anonymous and untraceable, and compares with existing DAA schemes in the aspects of security and efficiency.

We show that any language L in NP has a non-interactive zero-knowledge proof system which uses Θ(log(1/s) + n ∈) random bits, where s is the soundness error, n the length of the input and e can be any constant > 0. In order to achieve this result, we formulate and investigate the problem of randomness-efficient error reduction for non-interactive zero-knowledge proofs, which generalizes the analogue and well-studied problem for BPP computation.

Non-interactive zero-knowledge proof or argument (NIZK) systems are widely used in many security sensitive applications to enhance computation integrity, privacy and scalability. In such systems, a prover wants to convince one or more verifiers that the result of a public function is correctly computed without revealing the (potential) private input, such as the witness. In this work, we introduce a new notion, called scriptable SNARK, where the prover and verifier(s) can specify the function (or language instance) to be proven via a script. We formalize this notion in UC framework and provide a generic trusted hardware based solution. We then instantiate our solution in both SGX and Trustzone with Lua script engine. The system can be easily used by typical programmers without any cryptographic background. The benchmark result shows that our solution is better than all the known SNARK proof systems w.r.t. prover’s running time (1000 times faster), verifier’s running time, and the proof size. In addition, we also give a lightweight scriptable SNARK protocol for hardware with limited state, e.g., Θ ( λ ) bits. Finally, we show how the proposed scriptable SNARK can be readily deployed to solve many well-known problems in the blockchain context, e.g. verifier’s dilemma, fast joining for new players, etc.

Suppose we have a signature scheme for signing elements of message space \(\mathcal{M}_1\), but we need to sign messages from \(\mathcal{M}_2\). The traditional approach of applying a collision resistant hash function from \(\mathcal{M}_1\) to \(\mathcal{M}_2\) can be inconvenient when the signature scheme is used within more complex protocols, for example if we want to prove knowledge of a signature. Here, we present an alternative approach in which we can combine a signature for \(\mathcal{M}_1\), a pairwise independent hash function with key space \(\mathcal{M}_1\) and message space \(\mathcal{M}_2\), and a non-interactive zero knowledge proof system to obtain a signature scheme for message space \(\mathcal{M}_2\). This transform also removes any dependence on state in the signature for \(\mathcal{M}_1\).As a result of our transformation we obtain a new signature scheme for signing a vector of group elements that is based only on the decisional linear assumption (DLIN). Moreover, the public keys and signatures of our scheme consist of group elements only, and a signature is verified by evaluating a set of pairing-product equations, so the result is a structure-preserving signature. In combination with the Groth-Sahai proof system, such a signature scheme is an ideal building block for many privacy-enhancing protocols.

Privacy concerns of smart contracts are a major roadblock preventing their wider adoption. A promising approach to protect private data is hiding it with cryptographic primitives and then enforcing correctness of state updates by Non-Interactive Zero-Knowledge (NIZK) proofs. Unfortunately, NIZK statements are less expressive than smart contracts, forcing developers to keep some functionality in the contract. This results in scattered logic, split across contract code and NIZK statements, with unclear privacy guarantees. To address these problems, we present the zkay language, which introduces privacy types defining owners of private values. zkay contracts are statically type checked to (i) ensure they are realizable using NIZK proofs and (ii) prevent unintended information leaks. Moreover, the logic of zkay contracts is easy to follow by just ignoring privacy types. To enforce zkay contracts, we automatically transform them into contracts equivalent in terms of privacy and functionality, yet executable on public blockchains. We evaluated our approach on a proof-of-concept implementation generating Solidity contracts and implemented 10 interesting example contracts in zkay. Our results indicate that zkay is practical: On-chain cost for executing the transformed contracts is around 1M gas per transaction (~0.50US$) and off-chain cost is moderate.

The Fiat-Shamir (FS) transform is a well known and widely used technique to convert any constant-round public-coin honest-verifier zero-knowledge (HVZK) proof or argument system \(\mathsf {HVZK}=(\mathcal {P},\mathcal {V})\) in a non-interactive zero-knowledge (NIZK) argument system

A redactable signature scheme (\({\mathsf {RSS}}\)) allows removing parts of a signed message by any party without invalidating the respective signature. State-of-the-art constructions thereby focus on messages represented by one specific data-structure, e.g., lists, sets or trees, and adjust the security model accordingly. To overcome the necessity for this myriad of models, we present a general framework covering arbitrary data-structures and even more sophisticated possibilities. For example, we cover fixed elements which must not be redactable and dependencies between elements. Moreover, we introduce the notion of designated redactors, i.e., the signer can give some extra information to selected entities which become redactors. In practice, this often allows to obtain more efficient schemes. We then present two \(\mathsf {RSS}\)s; one for sets and one for lists, both constructed from any EUF-CMA secure signature scheme and indistinguishable cryptographic accumulators in a black-box way and show how the concept of designated redactors can be used to increase the efficiency of these schemes. Finally, we present a black-box construction of a designated redactor \(\mathsf {RSS}\) by combining an \(\mathsf {RSS}\) for sets with non-interactive zero-knowledge proof systems. All the three constructions presented in this paper provide transparency, which is an important property, but quite hard to achieve, as we also conceal the length of the original message and the positions of the redactions.

E-petition has played an important role in health and politics that collects public opinions and requests a superior or an authority to take actions towards a health or political problem. However, this activity exposes the privacy of the signers who participate to express opinions. In this paper, we propose a privacy-preserving fine-grained e-petition system that supports attribute-based identity verification for signers, while protecting their privacy. By considering the target groups of signers in a specific health or political petition, an attribute policy is defined to ensure that only the signers with the attributes that satisfy the attribute policy can sign the petition. The fine-grained petition is better than the traditional e-petitions because it can improve the trustworthiness of the petition results via proactive signer selection. Moreover, the new petition system protects the identities of the signers by using the non-interactive zero-knowledge proof system, such that the signers are anonymous in signing petitions. In addition, the proposed petition system supports the tracing of double-signing, a cheating behavior that an anonymous signer can submit more than one signature in a petition without being detected. Finally, we prove that the proposed petition system achieves the desirable security properties, including anonymity, unforgeability, and traceability, and demonstrate that the system is efficient to be implemented on the mobile devices.

A non-interactive zero-knowledge proof system (NIZK) for a composite language (i.e., a language that combines distinct statements by AND or OR) is widely used in cryptographic applications. The (im)possibility of black-box constructions of such NIZKs has been studied so far. We extend it to the case of proving equality of a witness behind distinct statements that often plays a central role in cryptographic constructions as illustrated by the seminal work by Naor and Yung (STOC, 1990). In this paper, we investigate the impossibility of a black-box construction of NIZKs for equality of witnesses from those proving the validity of the witnesses. Concretely, we prove that there exists no black-box construction of NIZKs for equality of plaintexts in two distinct ciphertexts of public-key encryption schemes based on NIZKs for proving validity of ciphertexts. It implies that one has to exploit specific properties of the underlying encryption schemes or NIZKs to construct NIZKs for proving equality, and hence justifies existing constructions.

Malleability for cryptography is not necessarily an opportunity for attack; in many cases it is a potentially useful feature that can be exploited. In this work, we examine notions of malleability for non-interactive zero-knowledge (NIZK) proofs. We start by defining a malleable proof system, and then consider ways to meaningfully control the malleability of the proof system, as in many settings we would like to guarantee that only certain types of transformations can be performed. As our motivating application, we consider a shorter proof for verifiable shuffles. Our controlled-malleable proofs allow us for the first time to use one compact proof to prove the correctness of an entire multi-step shuffle. Each authority takes as input a set of encrypted votes and a controlled-malleable NIZK proof that these are a shuffle of the original encrypted votes submitted by the voters; it then permutes and re-randomizes these votes and updates the proof by exploiting its controlled malleability. As another application, we generically use controlled-malleable proofs to realize a strong notion of encryption security. Finally, we examine malleability in existing proof systems and observe that Groth-Sahai proofs are malleable. We then go beyond this observation by characterizing all the ways in which they are malleable, and use them to efficiently instantiate our generic constructions from above; this means we can instantiate our proofs and all their applications using only the Decision Linear (DLIN) assumption.

This paper presents zero knowledge proofs, their cryptographic significance and applications. It presents a basic classification: interactive and noninteractive zero knowledge proofs. It presents and compares three protocols of non-interactive zero knowledge proofs: ZK-SNARK, ZK-STARK and Bulletproofs. It presents the quadratic residue problem and proofs it with both interactive and non-interactive zero knowledge proofs. The non-interactive protocol used to prove the quadratic residue problem is ZK-SNARK. The proof is implemented in the Python programming language, using python-snark library.

In a non-interactive zero-knowledge (NIZK) proof, a prover can non-interactively convince a verifier of a statement without revealing any additional information. Thus far, numerous constructions of NIZKs have been provided in the common reference string (CRS) model (CRS-NIZK) from various assumptions, however, it still remains a long standing open problem to construct them from tools such as pairing-free groups or lattices. Recently, Kim and Wu (CRYPTO’18) made great progress regarding this problem and constructed the first lattice-based NIZK in a relaxed model called NIZKs in the preprocessing model (PP-NIZKs). In this model, there is a trusted statement-independent preprocessing phase where secret information are generated for the prover and verifier. Depending on whether those secret information can be made public, PP-NIZK captures CRS-NIZK, designated-verifier NIZK (DV-NIZK), and designated-prover NIZK (DP-NIZK) as special cases. It was left as an open problem by Kim and Wu whether we can construct such NIZKs from weak paring-free group assumptions such as DDH. As a further matter, all constructions of NIZKs from Diffie-Hellman (DH) type assumptions (regardless of whether it is over a paring-free or paring group) require the proof size to have a multiplicative-overhead \(|C| \cdot \mathsf {poly}(\kappa )\), where |C| is the size of the circuit that computes the \(\mathbf {NP}\) relation.

In this paper we propose a two-round authenticated key agreement protocol using non-interactive zero-knowledge (NIZK) proofs. Essentially, a NIZK proof allows a prover to convince a verifier about the correctness of a statement without interaction. An obvious application of this is in a two-round key agreement protocol where each participant proves non-interactively the knowledge of the private key and a shared secret. Our key agreement protocol is based on this idea and is secure assuming the hardness of the Diffie-Hellman problem (DHP) and another related problem in bilinear maps.

Non-interactive zero-knowledge (NIZK) proofs for algebraic relations in a group, such as the Groth-Sahai proofs, are an extremely powerful tool in pairing-based cryptography. A series of recent works focused on obtaining very efficient NIZK proofs for linear spaces in a weaker quasi-adaptive model. We revisit recent quasi-adaptive NIZK constructions, providing clean, simple, and improved constructions via a conceptually different approach inspired by recent developments in identity-based encryption. We then extend our techniques also to linearly homomorphic structure-preserving signatures, an object both of independent interest and with many applications.

Non-interactive zero-knowledge proofs (NIZKs) are a powerful cryptographic tool, with numerous potential applications. However, succinct NIZKs (e.g., zk-SNARK schemes) necessitate a trusted party to generate and publish some public parameters, to be used by all provers and verifiers. This party is trusted to correctly run a probabilistic algorithm (specified by the the proof system) that outputs the public parameters, and publish them, without leaking any other information (such as the internal randomness used by the algorithm), violating either requirement may allow malicious parties to produce convincing "proofs" of false statements. This trust requirement poses a serious impediment to deploying NIZKs in many applications, because a party that is trusted by all users of the envisioned system may simply not exist. In this work, we show how public parameters for a class of NIZKs can be generated by a multi-party protocol, such that if at least one of the parties is honest, then the result is secure (in both aforementioned senses) and can be subsequently used for generating and verifying numerous proofs without any further trust. We design and implement such a protocol, tailored to efficiently support the state-of-the-art NIZK constructions with short and easy-to-verify proofs (Parno et al. IEEE S&P '13, Ben-Sasson et al. USENIX Sec '14, Danezis et al., ASIACRYPT '14). Applications of our system include generating public parameters for systems such as Zero cash (Ben-Sasson et al. IEEE S&P '13) and the scalable zero-knowledge proof system of (Ben-Sasson et al. CRYPTO '14).