In recent years, the zero-knowledge succinct non-interactive argument of knowledge (zk-SNARK) have drawn significant attention as privacy-enhancing technologies in various domains. A post-quantum designated verifier type zk-SNARK for Boolean circuits was proposed by Gennaro et al. in ACM CCS '18. However, this scheme does not include arithmetic circuits and they claim that a construction of post zk-SNARKs for arithmetic circuits as open problem. In this paper we give answers to this problem. Our first proposal is based on the data structure used in Pinocchio, a previous study, and can be easily implemented using the existing Pinocchio-based systems. In our second proposal, which also employs QAPs, the zero-knowledge proof comprises three learning with errors (LWE) ciphertexts, and the size of the proof is smaller compared with that of the first proposal. Our second proposal is also more efficient than the first one or all other known post-quantum zk-SNARKs.In recent years, the zero-knowledge succinct non-interactive argument of knowledge (zk-SNARK) have drawn significant attention as privacy-enhancing technologies in various domains. A post-quantum designated verifier type zk-SNARK for Boolean circuits was proposed by Gennaro et al. in ACM CCS '18. However, this scheme does not include arithmetic circuits and they claim that a construction of post zk-SNARKs for arithmetic circuits as open problem. In this paper we give answers to this problem. Our first proposal is based on the data structure used in Pinocchio, a previous study, and can be easily implemented using the existing Pinocchio-based systems. In our second proposal, which also employs QAPs, the zero-knowledge proof comprises three learning with errors (LWE) ciphertexts, and the size of the proof is smaller compared with that of the first proposal. Our second proposal is also more efficient than the first one or all other known post-quantum zk-SNARKs.

International Journal of Information and Computer SecurityJan 1, 20240 citations

Shorter ZK-SNARKs from square span programs over ideal lattices

Xi Lin, Heyang Cao, Feng-Hao Liu +2 more

Abstract Zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs) are cryptographic protocols that offer efficient and privacy-preserving means of verifying NP language relations and have drawn considerable attention for their appealing applications, e.g., verifiable computation and anonymous payment protocol. Compared with the pre-quantum case, the practicability of this primitive in the post-quantum setting is still unsatisfactory, especially for the space complexity. To tackle this issue, this work seeks to enhance the efficiency and compactness of lattice-based zk-SNARKs, including proof length and common reference string (CRS) length. In this paper, we develop the framework of square span program-based SNARKs and design new zk-SNARKs over cyclotomic rings. Compared with previous works, our construction is without parallel repetition and achieves shorter proof and CRS lengths than previous lattice-based zk-SNARK schemes. Particularly, the proof length of our scheme is around $23.3\%$ 23.3 % smaller than the recent shortest lattice-based zk-SNARKs by Ishai et al. (in: Proceedings of the 2021 ACM SIGSAC conference on computer and communications security, pp 212–234, 2021), and the CRS length is $3.6\times$ 3.6 × smaller. Our constructions follow the framework of Gennaro et al. (in: Proceedings of the 2018 ACM SIGSAC conference on computer and communications security, pp 556–573, 2018), and adapt it to the ring setting by slightly modifying the knowledge assumptions. We develop concretely small constructions by using module-switching and key-switching procedures in a novel way.Abstract Zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs) are cryptographic protocols that offer efficient and privacy-preserving means of verifying NP language relations and have drawn considerable attention for their appealing applications, e.g., verifiable computation and anonymous payment protocol. Compared with the pre-quantum case, the practicability of this primitive in the post-quantum setting is still unsatisfactory, especially for the space complexity. To tackle this issue, this work seeks to enhance the efficiency and compactness of lattice-based zk-SNARKs, including proof length and common reference string (CRS) length. In this paper, we develop the framework of square span program-based SNARKs and design new zk-SNARKs over cyclotomic rings. Compared with previous works, our construction is without parallel repetition and achieves shorter proof and CRS lengths than previous lattice-based zk-SNARK schemes. Particularly, the proof length of our scheme is around $$23.3\%$$ 23.3 % smaller than the recent shortest lattice-based zk-SNARKs by Ishai et al. (in: Proceedings of the 2021 ACM SIGSAC conference on computer and communications security, pp 212–234, 2021), and the CRS length is $$3.6\times$$ 3.6 × smaller. Our constructions follow the framework of Gennaro et al. (in: Proceedings of the 2018 ACM SIGSAC conference on computer and communications security, pp 556–573, 2018), and adapt it to the ring setting by slightly modifying the knowledge assumptions. We develop concretely small constructions by using module-switching and key-switching procedures in a novel way.

CybersecurityMar 19, 20244 citations

Quantum-Enhanced Zero-Knowledge Proofs for zk Rollup Security in Web3 Ecosystem

Niketa Yadav, Gaurav Indra

With the emergence of quantum computing, traditional cryptographic methods used in blockchain systems face increasing risk. One such area of concern is the Layer-2 protocols zk-Rollups designed to improve scalability and privacy in platforms like Ethereum, which are heavily dependent on classical zero-knowledge proofs such as zk-SNARKs and zk-STARKs. These systems may be compromised by quantum algorithms. To address this, we propose a quantum-secure zk-Rollup model using Quantum Zero-Knowledge Proofs (QZKPs), implemented with IBM’s Qiskit simulator. The protocol uses quantum features like superposition and random basis selection to verify transactions without leaking private data. Simulation results confirm key properties: valid proofs are reliably accepted, while invalid ones are rejected. This demonstrates both the feasibility and future relevance of integrating QZKPs into blockchain systems for post-quantum security.With the emergence of quantum computing, traditional cryptographic methods used in blockchain systems face increasing risk. One such area of concern is the Layer-2 protocols zk-Rollups designed to improve scalability and privacy in platforms like Ethereum, which are heavily dependent on classical zero-knowledge proofs such as zk-SNARKs and zk-STARKs. These systems may be compromised by quantum algorithms. To address this, we propose a quantum-secure zk-Rollup model using Quantum Zero-Knowledge Proofs (QZKPs), implemented with IBM’s Qiskit simulator. The protocol uses quantum features like superposition and random basis selection to verify transactions without leaking private data. Simulation results confirm key properties: valid proofs are reliably accepted, while invalid ones are rejected. This demonstrates both the feasibility and future relevance of integrating QZKPs into blockchain systems for post-quantum security.

Aug 11, 20251 citation

A tale of SNARKs: quantum resilience, knowledge extractability and data privacy

Anca Nitulescu

The contributions detailed in this thesis focus on the design and the analysis of Succinct non-interactive arguments of knowledge, known as SNARKs. SNARKs enable a party with large computational resources to prove to a weaker party that a particular statement is true in an efficient way without further interaction and under a minimal communication requirement. Our results deal with three different aspects of SNARK protocols: the postquantum security of SNARKs, the composability of SNARKs with other cryptographic primitives and the confidentiality of the inputs in the computations verified by SNARKs. First, we propose a new framework that allows the instantiation of a quantumresilient SNARK scheme from lattice assumptions. We also study the notion of extractability that is part of the soundness definition for SNARKs. We remark some limitations of this definition and we address this problem, by introducing and studying a new notion, O-SNARKs. Finally, to achieve data privacy in delegated computation, we study the possibility of constructing SNARKs that enables verification of computations over encrypted data.The contributions detailed in this thesis focus on the design and the analysis of Succinct non-interactive arguments of knowledge, known as SNARKs. SNARKs enable a party with large computational resources to prove to a weaker party that a particular statement is true in an efficient way without further interaction and under a minimal communication requirement. Our results deal with three different aspects of SNARK protocols: the postquantum security of SNARKs, the composability of SNARKs with other cryptographic primitives and the confidentiality of the inputs in the computations verified by SNARKs. First, we propose a new framework that allows the instantiation of a quantumresilient SNARK scheme from lattice assumptions. We also study the notion of extractability that is part of the soundness definition for SNARKs. We remark some limitations of this definition and we address this problem, by introducing and studying a new notion, O-SNARKs. Finally, to achieve data privacy in delegated computation, we study the possibility of constructing SNARKs that enables verification of computations over encrypted data.

Apr 1, 20190 citations

Zero-Knowledge Proofs and String Commitments Withstanding Quantum Attacks

Ivan Damgård, Serge Fehr, Louis Salvail

The concept of zero-knowledge (ZK) has become of fundamental importance in cryptography. However, in a setting where entities are modeled by quantum computers, classical arguments for proving ZK fail to hold since, in the quantum setting, the concept of rewinding is not generally applicable. Moreover, known classical techniques that avoid rewinding have various shortcomings in the quantum setting. We propose new techniques for building quantum zero-knowledge (QZK) protocols, which remain secure even under (active) quantum attacks. We obtain computational QZK proofs and perfect QZK arguments for any NP language in the common reference string model. This is based on a general method converting an important class of classical honest-verifier ZK (HVZK) proofs into QZK proofs. This leads to quite practical protocols if the underlying HVZK proof is efficient. These are the first proof protocols enjoying these properties, in particular the first to achieve perfect QZK. As part of our construction, we propose a general framework for building unconditionally hiding (trapdoor) string commitment schemes, secure against quantum attacks, as well as concrete instantiations based on specific (believed to be) hard problems. This is of independent interest, as these are the first unconditionally hiding string commitment schemes withstanding quantum attacks. Finally, we give a partial answer to the question whether QZK is possible in the plain model. We propose a new notion of QZK, non-oblivious verifier QZK, which is strictly stronger than honest-verifier QZK but weaker than full QZK, and we show that this notion can be achieved by means of efficient (quantum) protocols.The concept of zero-knowledge (ZK) has become of fundamental importance in cryptography. However, in a setting where entities are modeled by quantum computers, classical arguments for proving ZK fail to hold since, in the quantum setting, the concept of rewinding is not generally applicable. Moreover, known classical techniques that avoid rewinding have various shortcomings in the quantum setting. We propose new techniques for building quantum zero-knowledge (QZK) protocols, which remain secure even under (active) quantum attacks. We obtain computational QZK proofs and perfect QZK arguments for any NP language in the common reference string model. This is based on a general method converting an important class of classical honest-verifier ZK (HVZK) proofs into QZK proofs. This leads to quite practical protocols if the underlying HVZK proof is efficient. These are the first proof protocols enjoying these properties, in particular the first to achieve perfect QZK. As part of our construction, we propose a general framework for building unconditionally hiding (trapdoor) string commitment schemes, secure against quantum attacks, as well as concrete instantiations based on specific (believed to be) hard problems. This is of independent interest, as these are the first unconditionally hiding string commitment schemes withstanding quantum attacks. Finally, we give a partial answer to the question whether QZK is possible in the plain model. We propose a new notion of QZK, non-oblivious verifier QZK, which is strictly stronger than honest-verifier QZK but weaker than full QZK, and we show that this notion can be achieved by means of efficient (quantum) protocols.

BRICS Report SeriesMay 11, 200410 citations

Non-interactive zero-knowledge arguments for QMA, with preprocessing

Andrea Coladangelo, Thomas Vidick, Tina Zhang

We initiate the study of non-interactive zero-knowledge (NIZK) arguments for languages in QMA. Our first main result is the following: if Learning With Errors (LWE) is hard for quantum computers, then any language in QMA has an NIZK argument with preprocessing. The preprocessing in our argument system consists of (i) the generation of a CRS and (ii) a single (instance-independent) quantum message from verifier to prover. The instance-dependent phase of our argument system involves only a single classical message from prover to verifier. Importantly, verification in our protocol is entirely classical, and the verifier needs not have quantum memory; its only quantum actions are in the preprocessing phase. Our second contribution is to extend the notion of a classical proof of knowledge to the quantum setting. We introduce the notions of arguments and proofs of quantum knowledge (AoQK/PoQK), and we show that our non-interactive argument system satisfies the definition of an AoQK. In particular, we explicitly construct an extractor which can recover a quantum witness from any prover which is successful in our protocol. Finally, we show that any language in QMA has an (interactive) proof of quantum knowledge.We initiate the study of non-interactive zero-knowledge (NIZK) arguments for languages in QMA. Our first main result is the following: if Learning With Errors (LWE) is hard for quantum computers, then any language in QMA has an NIZK argument with preprocessing. The preprocessing in our argument system consists of (i) the generation of a CRS and (ii) a single (instance-independent) quantum message from verifier to prover. The instance-dependent phase of our argument system involves only a single classical message from prover to verifier. Importantly, verification in our protocol is entirely classical, and the verifier needs not have quantum memory; its only quantum actions are in the preprocessing phase. Our second contribution is to extend the notion of a classical proof of knowledge to the quantum setting. We introduce the notions of arguments and proofs of quantum knowledge (AoQK/PoQK), and we show that our non-interactive argument system satisfies the definition of an AoQK. In particular, we explicitly construct an extractor which can recover a quantum witness from any prover which is successful in our protocol. Finally, we show that any language in QMA has an (interactive) proof of quantum knowledge.

arXiv:1911.07546Nov 18, 20195 citations

23.3\%

23.3 % smaller than the recent shortest lattice-based zk-SNARKs by Ishai et al. (in: Proceedings of the 2021 ACM SIGSAC conference on computer and communications security, pp 212–234, 2021), and the CRS length is

3.6\times

3.6 × smaller. Our constructions follow the framework of Gennaro et al. (in: Proceedings of the 2018 ACM SIGSAC conference on computer and communications security, pp 556–573, 2018), and adapt it to the ring setting by slightly modifying the knowledge assumptions. We develop concretely small constructions by using module-switching and key-switching procedures in a novel way.