Circuit-based zero-knowledge proofs have arose as a solution to the implementation of privacy in blockchain applications, and to current scalability problems that blockchains suffer from. The most efficient circuit-based zero-knowledge proofs use a pairing-friendly elliptic curve to generate and validate proofs. In particular, the circuits are built connecting wires that carry elements from a large prime field, whose order is determined by the number of elements of the pairing-friendly elliptic curve. In this context, it is important to generate an inner curve using this field, because it allows to create circuits that can verify public-key cryptography primitives, such as digital signatures and encryption schemes. To this purpose, in this article, we present a deterministic algorithm for generating twisted Edwards elliptic curves defined over a given prime field. We also provide an algorithm for checking the resilience of this type of curve against most common security attacks. Additionally, we use our algorithms to generate Baby Jubjub, a curve that can be used to implement elliptic-curve cryptography in circuits that can be validated in the Ethereum blockchain.
With the development of Internet technology, Internet technology provides a platform for charity to allow people to participate together. Currently, each charitable blood donation registration agency operates independently, donors need to fill in personal information multiple times to register and log in when they want to register with different institutions. The information is collected by various platforms, which increases the risk of personal information leakage. In order to protect the security of the personal information used by the user during registration, and relieve the user from the many troubles caused by the leakage of personal information. This article proposes a user registration and login method based on blockchain technology, through the construction of a alliance chain, the blockchain technology is integrated with the registration and login system. To ensure that the user’s information is kept confidential to the charity blood donation platform and use the mobile phone number to register in the alliance chain under the premise of hiding the user’s login application record, successful registration can log in to all platforms on the alliance chain. The alliance chain system issues unique identification numbers to the charitable blood donation platform nodes in the chain, and users use their accounts to post transactions to apply for login nodes, and generate zero-knowledge range proofs to hide login records during the application login process.
Adrián Silveira, Gustavo Betarte, Maximiliano Cristiá, Carlos Luna
Mimblewimble is a privacy-oriented cryptocurrency technology which provides security and scalability properties that distinguish it from other protocols of its kind. In previous work we have proposed an idealized model that captures the main features of the Mimblewimble protocol with a model-driven verification approach. In this work, we present an extension of our model to enable zero-knowledge proofs in order to prove that the transaction amount is in a certain range without revealing the value. Furthermore, we provide some security properties a range proof scheme should satisfy.
Abstract We present a new zero-knowledge succinct argument of knowledge (zkSNARK) scheme for Rank-1 Constraint Satisfaction (RICS), a widely deployed NP-complete language that generalizes arithmetic circuit satisfiability. By instantiating with different commitment schemes, we obtain several zkSNARKs where the verifier’s costs and the proof size range from O (log 2 N ) to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mi>O</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:msqrt><m:mi>N</m:mi></m:msqrt></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:math> O\left( {\sqrt N } \right) depending on the underlying polynomial commitment schemes when applied to an N -gate arithmetic circuit. All these schemes do not require a trusted setup. It is plausibly post-quantum secure when instantiated with a secure collision-resistant hash function. We report on experiments for evaluating the performance of our proposed system. For instance, for verifying a SHA-256 preimage (less than 23k AND gates) in zero-knowledge with 128 bits security, the proof size is less than 150kB and the verification time is less than 11ms, both competitive to existing systems.
In recent years, securely sharing personal information between two parties involves high risk. Most of the medical health records and financial transactions includes huge uncertainty while storing and retrieving from cloud for query processing. Blockchain is an open platform where each transaction is tampered proof. Various methods like zero knowledge proof or hashing methods used to hide the sensitive information from the real world. When associating blockchain with cloud this uncertainty is reduced. The user information is segregated into two categories sensitive and non-sensitive using linear regression method before processing in cloud. To improve security and increase privacy from various attacks, the sensitive part of data is encrypted using ECC and non- sensitive part of data is encrypted using RSA algorithm. Using Ethereum blockchain the policy of the user is verified and query processing is done. The performance of the model is compared with the existing techniques and results are evaluated using the classification error rate and performance of security against manual attacks.
5G heterogeneous network (HetNet) is a novel network topology that integrates various kinds of wireless access technologies such as 4G Long-Term Evolution (LTE), Wi-Fi, and so on. Despite greatly improving spectrum efficiency, it poses enormous challenges to spectrum e-auction. Firstly, due to high mobility, bidders may be interested in different spectrums in terms of time or geolocation. Secondly, one’s bidding value should be protected against rival bidders or adversaries to avoid vicious competition as well as privacy leakage. Thirdly, the ubiquitous HetNet requires a trustworthy distributed auction framework rather than a centralized auctioneer-based pattern. Aiming at overcoming these obstacles above, we proposed a blockchain-based combinatorial spectrum e-auction framework. Different from other blockchain-based solutions of using SGX to realize trust processing in the auction phase, we adopt Zether, a privacy-preserving smart contract, as the main building block. Besides, the bidding value is preserved from the beginning to the end, even though the time-consuming Paillier homomorphic encryption and garbled circuits are absent. We provide the auction security by leveraging <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi mathvariant="normal">Σ</a:mi> </a:math> -Bullets, a zero-knowledge proof mechanism. Theoretical analysis and extensive evaluation also indicate that our approach is better than the state-of-the-art works in terms of efficiency and effectiveness.
—Algebra is one of the important fields of mathematics. It concerns with the study and manipulation of mathematical symbols. It also concerns with the study of abstractions such as groups, rings, and fields. Due to the development of these abstractions, it is extended to consider other structures, such as vectors, matrices, and polynomials, which are non-numerical objects. Computer algebra is the implementation of algebraic methods as algorithms and computer programs. Recently, many algebraic cryptosystem protocols are based on non-commutative algebraic structures, such as authentication, key exchange, and encryptiondecryption processes are adopted. Cryptography is the science that aimed at sending the information through public channels in such a way that only an authorized recipient can read it. Ring theory is the most attractive category of algebra in the area of cryptography. In this paper, we employ the algebraic structure called skew -Armendariz rings to design a neoteric algorithm for zero knowledge proof. The proposed protocol is established and illustrated through numerical example, and its soundness and completeness are proved
Zero-knowledge succinct arguments of knowledge (zkSNARKs) enable efficient privacy-preserving proofs of membership for general NP languages. Our focus in this work is on post-quantum zkSNARKs, with a focus on minimizing proof size. Currently, there is a 1000x gap in the proof size between the best pre-quantum constructions and the best post-quantum ones. Here, we develop and implement new lattice-based zkSNARKs in the designated-verifier preprocessing model. With our construction, after an initial preprocessing step, a proof for an NP relation of size 2^20 is just over 16 KB. Our proofs are 10.3x shorter than previous post-quantum zkSNARKs for general NP languages. Compared to previous lattice-based zkSNARKs (also in the designated-verifier preprocessing model), we obtain a 42x reduction in proof size and a 60x reduction in the prover's running time, all while achieving a much higher level of soundness. Compared to the shortest pre-quantum zkSNARKs by Groth (Eurocrypt 2016), the proof size in our lattice-based construction is 131x longer, but both the prover and the verifier are faster (by 1.2x and 2.8x, respectively). Our construction follows the general blueprint of Bitansky et al. (TCC 2013) and Boneh et al. (Eurocrypt 2017) of combining a linear probabilistically checkable proof (linear PCP) together with a linear-only vector encryption scheme. We develop a concretely-efficient lattice-based instantiation of this compiler by considering quadratic extension fields of moderate characteristic and using linear-only vector encryption over rank-2 module lattices.
MPC-in-the-Head (MitH) is a general framework that enables constructing efficient zero-knowledge (ZK) protocols for NP relations from secure multiparty computation (MPC) protocols. In this paper we present the first machine-checked implementations of MitH.
A random beacon provides a continuous public source of randomness and its applications range from public lotteries to zero-knowledge proofs. Existing random beacon protocols sacrifice either the fault tolerance or the communication complexity for security, or ease of reconfigurability. This work overcomes the challenges with the existing works through a novel communication efficient combination of state machine replication and (Publicly) Verifiable Secret Sharing (PVSS/VSS).
We propose a new doubly efficient interactive proof protocol for general arithmetic circuits. The protocol generalizes the interactive proof for layered circuits proposed by Goldwasser, Kalai and Rothblum to arbitrary circuits, while preserving the optimal prover complexity that is strictly linear to the size of the circuits. The proof size remains succinct for low depth circuits and the verifier time is sublinear for structured circuits. We then construct a new zero knowledge argument scheme for general arithmetic circuits using our new interactive proof protocol together with polynomial commitments. Our key technique is a new sumcheck equation that reduces a claim about the output of one layer to claims about its input only, instead of claims about all the layers above which inevitably incurs an overhead proportional to the depth of the circuit. We developed efficient algorithms for the prover to run this sumcheck protocol and to combine multiple claims back into one in linear time in the size of the circuit. Not only does our new protocol achieve optimal prover complexity asymptotically, but it is also efficient in practice. Our experiments show that it only takes 0.3 seconds to generate the proof for a circuit with more than 600,000 gates, which is 13 times faster than the original interactive proof protocol on the corresponding layered circuit. The proof size is 208 kilobytes and the verifier time is 66 milliseconds. Our implementation can take general arithmetic circuits directly, without transforming them to layered circuits with a high overhead on the size of the circuit.
Nicholas Franzese, Jonathan Katz, Steve Lu, Rafail Ostrovsky · 6 authors
We show a constant-overhead interactive zero-knowledge (ZK) proof system for RAM programs, that is, a ZK proof in which the communication complexity as well as the running times of the prover and verifier scale linearly in the size of the memory N and the running time T of the underlying RAM program. Besides yielding an asymptotic improvement of prior work, our implementation gives concrete performance improvements for RAM-based ZK proofs. In particular, our implementation supports ZK proofs of private read/write accesses to 64~MB of memory (224 32-bit words) using only 34~bytes of communication per access, a more than 80x improvement compared to the recent BubbleRAM protocol. We also design a lightweight RISC CPU that can efficiently emulate the MIPS-I instruction set, and for which our ZK proof communicates only ~320 bytes per cycle, more than 10x less than the BubbleRAM CPU. In a 100 Mbps network, we can perform zero-knowledge executions of our CPU (with 64~MB of main memory and 4~MB of program memory) at a clock rate of 6.6 KHz.
Yun Li, Cun Ye, Yuguang Hu, Ivring Morpheus · 10 authors
Devising a fair-exchange protocol for digital goods has been an appealing line of research in the past decades. The Zero-Knowledge Contingent Payment (ZKCP) protocol first achieves fair exchange in a trustless manner with the aid of the Bitcoin network and zero-knowledge proofs. However, it incurs setup issues and substantial proving overhead, and has difficulties handling complicated validation of large-scale data. In this paper, we propose an improved solution ZKCPlus for practical and flexible fair exchange. ZKCPlus incorporates a new commit-and-prove non-interactive zero-knowledge (CP-NIZK) argument of knowledge under standard discrete logarithmic assumption, which is prover-efficient for data-parallel computations. With this argument we avoid the setup issues of ZKCP and reduce seller's proving overhead, more importantly enable the protocol to handle complicated data validations. We have implemented a prototype of ZKCPlus and built several applications atop it. We rework a ZKCP's classic application of trading sudoku solutions, and ZKCPlus achieves 21-67 times improvement in seller efficiency than ZKCP, with only milliseconds of setup time and 1 MB public parameters. In particular, our CP-NIZK argument shows an order of magnitude higher proving efficiency than the zkSNARK adopted by ZKCP. We also built a realistic application of trading trained CNN models. For a 3-layer CNN containing 8,620 parameters, it takes less than 1 second to prove and verify an inference computation, and also about 1 second to deliver the parameters, which is very promising for practical use.
Cyprien Delpech de Saint Guilhem, Emmanuela Orsini, Titouan Tanguy
This work introduces a new interactive oracle proof system based on the MPC-in-the-Head paradigm. To improve concrete efficiency and offer flexibility between computation time and communication size, a generic proof construction based on multi-round MPC protocols is proposed, instantiated with a specific protocol and implemented and compared to similar proof systems.
Zhiyong Fang, David Darais, Joseph P. Near, Yupeng Zhang
Static program analysis tools can automatically prove many useful properties of programs. However, using static analysis to prove to a third party that a program satisfies a property requires revealing the program's source code. We introduce the concept of zero-knowledge static analysis, in which the prover constructs a zero-knowledge proof about the outcome of the static analysis without revealing the program. We present novel zero-knowledge proof schemes for intra- and inter-procedural abstract interpretation. Our schemes are significantly more efficient than the naive translation of the corresponding static analysis algorithms using existing schemes. We evaluate our approach empirically on real and synthetic programs; with a pairing-based zero knowledge proof scheme as the backend, we are able to prove the control flow analysis on a 2,000-line program in 1,738s. The proof is only 128 bytes and the verification time is 1.4ms. With a transparent zero knowledge proof scheme based on discrete-log, we generate the proof for the tainting analysis on a 12,800-line program in 406 seconds, the proof size is 282 kilobytes, and the verification time is 66 seconds.
Zero-knowledge proof (ZKP) is a fundamental cryptographic primitive that allows a prover to convince a verifier of the validity of a statement without leaking any further information. As an efficient variant of ZKP, noninteractive zero-knowledge proof (NIZKP) adopting the Fiat-Shamir heuristic is essential to a wide spectrum of applications, such as federated learning, blockchain, and social networks. However, the heuristic is typically built upon the random oracle model that makes ideal assumptions about hash functions, which does not hold in reality and thus undermines the security of the protocol. Here, we present a quantum solution to the problem. Instead of resorting to a random oracle model, we implement a quantum randomness service. This service generates random numbers certified by the loophole-free Bell test and delivers them with postquantum cryptography (PQC) authentication. By employing this service, we conceive and implement NIZKP of the three-coloring problem. By bridging together three prominent research themes, quantum nonlocality, PQC, and ZKP, we anticipate this work to inspire more innovative applications that combine quantum information science and the cryptography field.
Zero-knowledge (ZK) proofs with an optimal memory footprint have attracted a lot of attention, because such protocols can easily prove very large computation with a small memory requirement. Such ZK protocol only needs O(M) memory for both parties, where M is the memory required to verify the statement in the clear. In this paper, we propose several new constant-round ZK protocols in this setting, which improve the concrete efficiency and, at the same time, enable sublinear amortized communication for circuits with some notion of relaxed uniformity. In the circuit-based model, where the computation is represented as a circuit over a field, our ZK protocol achieves a communication complexity of 1 field element per non-linear gate for any field size while keeping the computation very cheap. We implemented our protocol, which shows extremely high efficiency and affordability. Compared to the previous best-known implementation, we achieve 6x--7x improvement in computation and 3x--7x improvement in communication. When running on intro-level AWS instances, our protocol only needs one US dollar to prove one trillion AND gates (or 2.5 US dollars for one trillion multiplication gates over a 61-bit field). In the setting where part of the computation can be represented as a set of polynomials with a "degree-separated" format, we can achieve communication sublinear to the polynomial size: the communication only depends on the total number of distinct variables in all the polynomials and the highest degree of all polynomials, independent of the number of multiplications to compute all polynomials. Using the improved ZK protocol, we can prove matrix multiplication with communication proportional to the input size, rather than the number of multiplications. Proving the multiplication of two 1024 x 1024 matrices, our implementation, with one thread and 1 GB of memory, only needs 10 seconds and communicates 25 MB.
Deep learning techniques with neural networks are developing prominently in recent years and have been deployed in numerous applications. Despite their great success, in many scenarios it is important for the users to validate that the inferences are truly computed by legitimate neural networks with high accuracy, which is referred to as the integrity of machine learning predictions. To address this issue, in this paper, we propose zkCNN, a zero knowledge proof scheme for convolutional neural networks (CNN). The scheme allows the owner of the CNN model to prove to others that the prediction of a data sample is indeed calculated by the model, without leaking any information about the model itself. Our scheme can also be generalized to prove the accuracy of a secret CNN model on a public dataset.
Carsten Baum, Lennart Braun, Alexander Munch-Hansen, Benoît Razet · 5 authors
Zero-knowledge proofs are highly flexible cryptographic protocols that are an important building block for many secure systems. Typically, these are defined with respect to statements that are formulated as arithmetic operations over a fixed finite field. This inflexibility is a disadvantage when it comes to complex programs, as some fields are more amenable to express certain operations than others. At the same time, there do not seem to be many proofs with a programming model similar to those found in modern computer architectures that perform arithmetic with 32 or 64 bit integers.
Federated machine learning (FL) allows to collectively train models on sensitive data as only the clients' models and not their training data need to be shared. However, despite the attention that research on FL has drawn, the concept still lacks broad adoption in practice. One of the key reasons is the great challenge to implement FL systems that simultaneously achieve fairness, integrity, and privacy preservation for all participating clients. To contribute to solving this issue, our paper suggests a FL system that incorporates blockchain technology, local differential privacy, and zero-knowledge proofs. Our implementation of a proof-of-concept with multiple linear regression illustrates that these state-of-the-art technologies can be combined to a FL system that aligns economic incentives, trust, and confidentiality requirements in a scalable and transparent system.
Aiming at the problems of large ciphertext size and low efficiency in the current secure multi-party computation (SMC) protocol based on fully homomorphic encryption (FHE), the paper proves that the fully homomorphic encryption scheme that supports multi-bit encryption proposed by Chen Li et al. satisfies the key homomorphism. Based on this scheme and threshold decryption, a three-round, interactive, leveled, secure multi-party computation protocol under the Common Random String (CRS) model is designed. The protocol is proved to be safe under the semi-honest model and the semi-malicious model. From the non-interactive zero-knowledge proof, it can be concluded that the protocol is also safe under the malicious model. Its security can be attributed to the Decisional Learning With Errors (DLWE) and a variant of this problem (some-are-errorless LWE). Compared with the existing secure multi-party computation protocol based on fully homomorphic encryption under the CRS model, the ciphertext size of this protocol is smaller, the efficiency is higher, the storage overhead is smaller, and the overall performance is better than the existing protocol.
The labeling of electricity is considered an important mechanism to differentiate renewable power generation and, thus, to incentivize the expansion of green energy. However, today's systems for documenting and trading green energy certificates suffer from multiple challenges. These could be addressed by a digital solution that holistically collects and processes production and consumption data. Blockchain-based architectures have repeatedly been suggested for this purpose since they can provide transparency and can likely be accepted by a broad group of stakeholders. Yet, there are significant scalability and privacy issues of a blockchain-based approach for storing and processing fine-grained production and consumption data. In this paper, we propose and discuss a potential solution that levers succinct cryptographic zero-knowledge proofs to balance the required level of transparency and privacy while at the same time providing a high degree of scalability.