Zero-knowledge Proofs Research Articles

PriorMSM: An Efficient Acceleration Architecture for Multi-Scalar Multiplication

Multi-Scalar Multiplication (MSM) is a computationally intensive task that operates on elliptic curves based on GF(P) . It is commonly used in zero-knowledge proof (ZKP), where it accounts for a significant portion of the computation time required for proof generation. In this article, we present PriorMSM, an efficient acceleration architecture for MSM. We propose a Priority-Based Scheduling Mechanism (PBSM) based on a multi-FIFO and multi-bank architecture to accelerate the implementation of MSM. By increasing the pairing success rate of internal points, PBSM reduces the number of bubbles in the pipeline of point addition (PADD), consequently improving the data throughput of the pipeline. We also introduce an advanced parallel bucket aggregation algorithm, leveraging PADD’s fully pipelined characteristics to significantly accelerate the implementation of bucket aggregation. We perform a sensitivity analysis on the crucial parameter of window size in MSM. The results indicate that the window size of the MSM significantly impacts its latency. Area-Time Product (ATP) metric is introduced to guide the selection of the optimal window size, balancing the performance and cost for practical applications of subsequent MSM implementations. PriorMSM is evaluated using the TSMC 28 nm process. It achieves a maximum speedup of 10.9× compared to the previous custom hardware implementations and a maximum speedup of 3.9× compared to the GPU implementations.

Research and Application Analysis of Key Technologies of Zero-Knowledge Proof under the Background of Blockchain

Research and Application Analysis of Key Technologies of Zero-Knowledge Proof under the Background of Blockchain

Smart Contracts Management: The Interplay of Data Privacy and Blockchain for Secure and Efficient Real Estate Transactions

The digital transformation of the real estate industry is being significantly influenced by blockchain technology and smart contracts, which promise enhanced efficiency, transparency, and security in transactions. This study aims to develop a secure and efficient smart contract management protocol that balances the benefits of blockchain with robust data privacy practices. The methodology involves descriptive analytics of transaction data from the Ethereum blockchain, feasibility studies using synthetic transaction data, and a regulatory compliance analysis to map the impact of different regions' regulations on blockchain adoption in real estate. The findings reveal that while smart contracts can automate various processes and reduce reliance on intermediaries, challenges related to data privacy and regulatory compliance persist. Higher privacy features in smart contracts are associated with increased execution costs, indicating a trade-off between privacy and cost efficiency. Smart contracts with privacy level 3 had an execution cost of 0.025 ETH, compared to those with privacy level 1 at 0.02 ETH. Integrating permissioned blockchains and zero-knowledge proofs offers a promising solution, though their complexity limits broader adoption. Zero-knowledge proofs maintained high privacy (achieving privacy levels of up to 0.76) at a reasonable computational cost (proof generation time of 1.9 seconds). Thus, the integration of permissioned blockchains and zero-knowledge proofs offers a promising pathway to address these challenges. However, the complexity of these techniques requires specialized knowledge, limiting broader adoption. The study concludes with recommendations to develop specialized training programs, collaborate on regulatory frameworks, invest in advanced cryptographic research, and implement targeted strategies to overcome adoption barriers. These efforts will contribute to the digital transformation of asset management, fostering innovation and enhancing the overall efficiency of real estate transactions.

Evaluating the Efficiency of zk-SNARK, zk-STARK, and Bulletproof in Real-World Scenarios: A Benchmark Study

This study builds on our previous systematic literature review (SLR) that assessed the applications and performance of zk-SNARK, zk-STARK, and Bulletproof non-interactive zero-knowledge proof (NIZKP) protocols. To address the identified research gaps, we designed and implemented a benchmark comparing these three protocols using a dynamic minimized multiplicative complexity (MiMC) hash application. We evaluated performance across four general-purpose programming libraries and two programming languages. Our results show that zk-SNARK produced the smallest proofs, while zk-STARK generated the largest. In terms of proof generation and verification times, zk-STARK was the fastest, and Bulletproof was the slowest. Interestingly, zk-SNARK proofs verified marginally faster than zk-STARK, contrary to other findings. These insights enhance our understanding of the functionality, security, and performance of NIZKP protocols, providing valuable guidance for selecting the most suitable protocol for specific applications.

Towards zero knowledge argument for double discrete logarithm with constant cost

Given that the Schnorr's protocol for Discrete Logarithm (DLOG) exchanges three messages, it is an interesting problem whether a constant round zero-knowledge protocol exists for the Double Discrete Logarithm problem (DDLOG), i.e., to demonstrate the knowledge of a secret witness x in ghx. In this paper, we show that it exists for a fragment of DDLOG with two restrictions: (1) The outer group of DDLOG supports bilinear pairing, and it needs a trusted set-up for common reference string (CRS). (2) x<t where t is the size of KZG commitment key in CRS. The protocol is zero knowledge and constant round, with O(1) complexity for prover and verifier, regardless of the desired security strength. The contributions of the work are mainly theoretical due to its restrictions and concrete performance.

Secure Processing and Distribution of Data Managed on Private InterPlanetary File System Using Zero-Knowledge Proofs

In this study, a new data-sharing method is proposed that uses a private InterPlanetary File System—a decentralized storage system operated within a closed network—to distribute data to external entities while making its authenticity verifiable. Among the two operational modes of IPFS, public and private, this study focuses on the method for using private IPFS. Private IPFS is not open to the general public; although it poses a risk of data tampering when distributing data to external parties, the proposed method ensures the authenticity of the received data. In particular, this method applies a type of zero-knowledge proof, namely, the Groth16 protocol of zk-SNARKs, to ensure that the data corresponds to the content identifier in a private IPFS. Moreover, the recipient’s name is embedded into the distributed data to prevent unauthorized secondary distribution. Experiments confirmed the effectiveness of the proposed method for an image data size of up to 120 × 120 pixels. In future studies, the proposed method will be applied to larger and more diverse data types.

NP-Completeness and Physical Zero-Knowledge Proofs for Sumplete, a Puzzle Generated by ChatGPT

Sumplete is a logic puzzle generated by ChatGPT in March 2023. The puzzle consists of a rectangular grid, with each cell containing an integer. Each row and column also has an integer called target value assigned to it. The objective of this puzzle is to cross out some numbers in the grid such that the sum of uncrossed numbers in each row and column is equal to the corresponding target value. In this paper, we prove that Sumplete is NP-complete. We also propose a physical zero-knowledge proof protocol for the puzzle using physical cards.

Algebraic Structures and Their Applications in Modern Cryptography

Modern cryptography relies heavily on the principles of algebraic structures to ensure the security and integrity of data. This paper explores the fundamental algebraic structures that underpin contemporary cryptographic systems, including groups, rings, fields, and lattices. We provide a detailed examination of how these structures are employed in various cryptographic algorithms and protocols, such as public-key cryptography, digital signatures, and hash functions. an overview of basic algebraic concepts and their properties, followed by an in-depth analysis of their applications in cryptographic schemes. For instance, the use of elliptic curve groups in Elliptic Curve Cryptography (ECC) offers enhanced security with smaller key sizes compared to traditional systems like RSA. Similarly, lattice-based cryptography presents promising solutions for post-quantum security, leveraging the hardness of lattice problems to resist attacks by quantum computers. the role of algebraic structures in the development of advanced cryptographic techniques, such as homomorphic encryption, which allows computations on encrypted data without decryption, and zero-knowledge proofs, which enable the verification of information without revealing the information itself. Through these examples, we illustrate the critical importance of algebraic structures in achieving robust and efficient cryptographic systems.

Research on ZKP Algorithm of Data Asset Security and Privacy Protection Based on Blockchain Technology

Zero Knowledge Proof (ZKP) is a very effective method of preserving privacy as it hides the most confidential information throughout the transaction. In this paper, we present a security and privacy-preserving approach for blockchain that relies on account and multi-data asset models using the Zero Knowledge Proof (ZKP) mechanism. We provide options for transferring data assets and detecting duplicate expenditures, and we also develop transaction structures, anonymised addresses and anonymised metadata for the data assets. To create and validate the ZKP, we use the zk-SNARKs algorithm and specify validation criteria for masked transactions, and finally conduct experimental tests to validate it. Creating better algorithms for ZKP will be the focus of our future efforts.

Utilizing Blockchain Technology to Modernize Police Operations: Ensuring Security, Transparency, and Efficiency

This article explores the transformative potential of blockchain technology in police operations, focusing on enhancing security, transparency, and efficiency. Initially recognized for its role in cryptocurrencies, blockchain's inherent immutability and distributed ledger features are now being leveraged to revolutionize data management within law enforcement. The technology ensures the secure, auditable, and tamper-resistant recording of all transactions and data related to police operations, from investigation records to evidence management, ensuring accessibility only to authorized personnel. Key areas examined include the enhancement of security and transparency through blockchain's immutable and distributed records, ensuring data integrity and evidence management by providing an unalterable record of all transactions and data, and improving interoperability among various police departments and agencies. The integration of advanced technologies like Zero-Knowledge Proofs and Layer 2 solutions further strengthens privacy protection and operational speed, contributing to a more efficient and reliable police information system. Practical implementations in police departments worldwide, such as Delhi Police's blockchain initiative for evidence tracking and Dubai Police's collaboration with Cardano Blockchain for secure data sharing, demonstrate significant improvements in transparency, integrity, and security of processes. The article concludes by addressing the ongoing challenges of scalability, interoperability, and privacy, emphasizing the need for continued development and standardization to fully realize blockchain's benefits in modernizing police operations. By evaluating these aspects, the article highlights blockchain's critical role in the digital transformation of law enforcement, promoting greater public trust and cooperation among police units at both national and international levels.

Physical Zero-Knowledge Proof Protocols for Topswops and Botdrops

Suppose that a sequence of n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{n}}$$\\end{document} cards, numbered 1 to n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{n}}$$\\end{document}, is placed face up in random order. Let k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{k}}$$\\end{document} be the number on the first card in the sequence. Then take the first k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{k}}$$\\end{document} cards from the sequence, rearrange that subsequence of k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{k}}$$\\end{document} cards in reverse order, and return them to the original sequence. Repeat this prefix reversal until the number on the first card in the sequence becomes 1. This is a one-player card game called Topswops. The computational complexity of Topswops has not been thoroughly investigated. For example, letting f(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{f}}({\\varvec{n}})$$\\end{document} denote the maximum number of prefix reversals for Topswops with n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{n}}$$\\end{document} cards, values of f(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{f}}({\\varvec{n}})$$\\end{document} for n≥20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{n}}\\ge 20$$\\end{document} remain unknown. In general, there is no known efficient algorithm for finding an initial sequence of n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{n}}$$\\end{document} cards that requires exactly ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} prefix reversals for any integers n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{n}}$$\\end{document} and ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\\ell }}$$\\end{document}. In this paper, using a deck of cards, we propose a physical zero-knowledge proof protocol that allows a prover to convince a verifier that the prover knows an initial sequence of n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{n}}$$\\end{document} cards that requires ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\\ell }}$$\\end{document} prefix reversals without leaking knowledge of that sequence. We also deal with Botdrops, a variant of Topswops.

Uni/multi variate polynomial embeddings for zkSNARKs

AbstractA zero-knowledge proof is a cryptographic primitive that enables a prover to convince a verifier the validity of a mathematical statement (an NP statement) without revealing any secret inputs to the verifier. A special case, called zero-knowledge Succinct Non-interactive ARgument of Knowledge (zkSNARK) is particularly designed for arithmetic circuit proof systems which have important applications in blockchain privacy. The major computations in this type of zkSNARK proofs with post-quantum security are polynomial evaluations and Lagrange interpolations over finite fields. Given a sequence over a finite field, in the field of coding and sequences research, we understand that there are two representations of the sequence, one is a univariate polynomial and the other, a multivariate polynomial. This is exactly what is done in those zero-knowledge proof systems to transform the proof of a R1CS relation to evaluate uni/multi variate polynomials at some random points in the finite field. In this paper, we present a comparative analysis on how to convert a rank 1 constrained satisfiability (R1CS) system (more general than a circuit system) into a polynomial equality and provide analysis on the concrete complexities of provers, proof sizes and verifiers. We use two concrete zkSNARK schemes, i.e., Polaris, univariate polynomial encodings and Spartan, multivariate polynomial encodings, as examples to show our analysis. Secondly, we propose to select interpolating sets as subfields instead of affine spaces of a large field for Lagrange interpolation. This new method has improved the performance of R1CS encodings largely. We comment that post-quantum secure zkSNARKs yield post-quantum digital signatures with security only depending on symmetric-key schemes. Some open problems are proposed at the end of the paper.

Physical Zero-Knowledge Proof for Sukoro

A zero-knowledge proof protocol is a cryptographic protocol in which a prover, who knows the witness to a statement, can convince a verifier that the statement is true without revealing any information about the witness. Although zero-knowledge proof protocols are typically executed on electronic computers, there is a line of research to design zero-knowledge proof protocols based on physical objects (e.g., a deck of cards). This is called physical zero-knowledge proof. In this paper, we construct a physical zero-knowledge proof protocol for a logical puzzle called Sukoro. Sukoro has many cells on the puzzle board, like Sudoku, where each cell must be empty or filled with a number from one to four, and each number must match the number of adjacent filled cells, and the same numbers must not be adjacent to each other. In addition, it has a rule that all filled cells must be connected, which is called the connectivity condition. Although some existing protocols deal with the connectivity condition, all existing methods are interactive, which requires the prover’s knowledge to determine how the cards are manipulated during the execution of the protocols. In this paper, we give a new method for verifying the connectivity condition in the non-interactive setting, which means that the protocol can be executed without the prover’s knowledge, and construct a physical zero-knowledge proof protocol for Sukoro.

LiteHash: Hash Functions for Resource-Constrained Hardware

The global paradigm shift towards edge computing has led to a growing demand for efficient integrity verification. Hash functions are one-way algorithms which act as a zero-knowledge proof of a datum’s contents. However, it is infeasible to compute hashes on devices with limited processing power and memory. Hence, we propose four novel LiteHash functions which are architecturally similar to SHA-512 yet simpler. By using various approximation techniques, our implementations reduce the computational costs of digesting a message into a hash. On validating our proposed designs using the NIST PRNG Test Suite, we observe SHA-512 equivalent cryptographic security while satisfying all desired hash function property requirements. We observe a minimum of 9.41% reduction in area, 20.47% reduction in power and 22.05% increase in throughput. Our designs offer a throughput of upto 2 Gbps while reducing area and power by a maximum of 16.86% and 32.48% respectively. LiteHash functions also support the computation of the entire SHA-2 family of hash functions (SHA-224/256/384/512) with minor architectural modifications.

Zero-Knowledge Proofs and Their Role within the Blockchain

In each issue of Communications , we publish selected posts or excerpts from the many blogs on our website. The views expressed by bloggers are their own and not necessarily held by Communications or the Association for Computing Machinery. Read more blogs and join the discussion at https://cacm.acm.org/blog. https://cacm.acm.org/blog An examination of the intricate world of zero-knowledge proofs and how they may be used to enhance blockchain security and privacy.

AnoPay: Anonymous Payment for Vehicle Parking With Updatable Credential

Many existing anonymous parking payment schemes lack high efficiency and flexibility. For instance, the calculation and communication costs involved in payment may linearly increase with the payment amount. In this paper, we propose an anonymous payment system (dubbed AnoPay) for vehicle parking, which leverages updatable attribute-based anonymous credentials and efficient zero-knowledge proof (ZKP) to achieve user anonymity and constant overhead for parking fee payment. To further improve the efficiency, we design a secure parking fee aggregation protocol based on linear homomorphic encryption to aggregate parking transactions, where the amount of each parking transaction is hidden and the privacy of the parking lot in terms of its revenue is guaranteed. AnoPay achieves both unlinkability and accountability, malicious payments can be efficiently traced when it is necessary. We provide a security model and rigorous proof for each security property of AnoPay. Extensive experiments and comparisons demonstrate the efficiency and practicality of the system.

Compact Issuer-Hiding Authentication, Application to Anonymous Credential

Anonymous credentials are cryptographic mechanisms enabling users to authenticate themselves with a fine-grained control on the information they leak in the process. They have been the topic of countless papers which have improved the performance of such mechanisms or proposed new schemes able to prove ever-more complex statements about the attributes certified by those credentials. However, although these papers have studied in depth the problem of the information leaked by the credential and/or the attributes, almost all of them have surprisingly overlooked the information one may infer from the knowledge of the credential issuer. In this paper we address this problem by showing how one can efficiently hide the actual issuer of a credential within a set of potential issuers. The novelty of our work is that we do not resort to zero-knowledge proofs but instead we show how one can tweak Pointcheval-Sanders signatures to achieve this issuer-hiding property in a compact way. This results in an efficient anonymous credential system that indeed provides a complete control of the information leaked in the authentication process. Our construction is moreover modular and can then fit a wide spectrum of applications, notably for Self-Sovereign Identity (SSI) systems.

BRON: A blockchained framework for privacy information retrieval in human resource management

The correctness and the true validated data in Human Resource Management (HRM) are important for organizations as the data plays an impactful role in recruiting, developing, and retaining a skilled workforce. On one hand, the validated data in an organization helps in recruiting legitimate skillful employees; on the other hand, keeping the employee's data safe and maintaining privacy laws such as compliance with the General Data Protection Regulation (GDPR) is also an organization's responsibility. Besides, transparency in human resource management operations is crucial because it promotes trust and fairness within an organization. The present HRM systems are centralized in nature and their verifiable credential system is ineffective; this leads to the intentions of internal data sabotage or internal threats. Besides, the organizations' biases also become more prominent.In this paper, we address the above-mentioned problems with a blockchain framework for HRM to utilize the privacy of data access through a Privacy Information Retrieval (PIR) process. To be specific, our proposed framework called Blockchained piR of resOurces as humaN (BRON), is the first blockchain framework to show an effective mechanism to access data from organizations globally without hampering privacy. BRON uses a generalized user registration process to use the services of data access and in the background, it uses Zero-Knowledge Proofs (ZKPs) for global verification and PIR for privacy-based data retrieval. More specifically, credential verification and ZKP-based PIR are the highlights of our proposed BRON. Another interesting aspect of BRON is the use of Proof-of-Authority (PoA) to validate the anonymity and unlinkability of any HR operation. Finally, BRON has also contributed with a smart contract to incentivize the employees. BRON is very generic and easily be customizable as per the HR requirements. We run a set of experiments on BRON and observe that it is successful in providing privacy-assured data access and decentralized human resource data management. Overall, BRON provides 30% reduced latency and 35% better throughput as compared to the existing blockchain solutions in the direction of HRM.

VeriCNN: Integrity verification of large-scale CNN training process based on zk-SNARK

Cloud environments are frequently employed to train Convolutional Neural Networks (CNNs). On the other hand, the training procedure is exposed to potential integrity threats due to the cloud’s lack of trustworthiness. Although there are now available methods for zero-knowledge proof-based training integrity verification for small models, the low-proof performance of large-scale CNNs like LeNet-5 and VGG16 makes this task challenging. To solve the training integrity verification of large-scale CNNs, this research suggests a technique called VeriCNN. Particularly, VeriCNN builds proof of integrity for CNN training using zk-SNARK. It adopts the Winograd algorithm to optimize convolution operations and introduces a high-probability matrix multiplication to optimize zero-knowledge proofs. Experimental results demonstrate the effectiveness and practicality of VeriCNN in verifying the training integrity of large-scale models. For instance, VeriCNN can generate a training integrity proof for the LeNet-5 model in just 25 s and for the VGG16 model in 390 s.

Printing Protocol: Physical ZKPs for Decomposition Puzzles

Decomposition puzzles are pencil-and-paper logic puzzles that involve partitioning a rectangular grid into several regions to satisfy certain rules. In this paper, we construct a generic card-based protocol called printing protocol, which can be used to physically verify solutions of decompositon puzzles. We apply the printing protocol to develop card-based zero-knowledge proof protocols for two such puzzles: Five Cells and Meadows. These protocols allow a prover to physically show that he/she knows solutions of the puzzles without revealing them.