François Le Gall, Yupan Liu, Harumichi Nishimura, Qisheng Wang
We introduce two models of space-bounded quantum interactive proof systems, QIPL and QIP_{U}L. The QIP_{U}L model, a space-bounded variant of quantum interactive proofs (QIP) introduced by Watrous (CC 2003) and Kitaev and Watrous (STOC 2000), restricts verifier actions to unitary circuits. In contrast, QIPL allows logarithmically many pinching intermediate measurements per verifier action, making it the weakest model that encompasses the classical model of Condon and Ladner (JCSS 1995). We characterize the computational power of QIPL and QIP_{U}L. When the message number m is polynomially bounded, QIP_{U}L ⊊ QIPL unless P = NP: - QIPL^HC, a subclass of QIPL defined by a high-concentration condition on yes instances, exactly characterizes NP. - QIP_{U}L is contained in P and contains SAC¹ ∪ BQL, where SAC¹ denotes problems solvable by classical logarithmic-depth, semi-unbounded fan-in circuits. However, this distinction vanishes when m is constant. Our results further indicate that (pinching) intermediate measurements uniquely impact space-bounded quantum interactive proofs, unlike in space-bounded quantum computation, where BQL = BQ_{U}L. We also introduce space-bounded unitary quantum statistical zero-knowledge (QSZK_{U}L), a specific form of QIP_{U}L proof systems with statistical zero-knowledge against any verifier. This class is a space-bounded variant of quantum statistical zero-knowledge (QSZK) defined by Watrous (SICOMP 2009). We prove that QSZK_{U}L = BQL, implying that the statistical zero-knowledge property negates the computational advantage typically gained from the interaction.
This paper presents a comprehensive analysis of storage proofs in the Ethereum ecosystem, examining their role in addressing historical and cross-chain state access challenges. We systematically review existing approaches to historical state verification, comparing Merkle Mountain Range (MMR) and Merkle-Patricia trie (MPT) architectures. An analysis involves their respective performance characteristics within zero-knowledge contexts, where performance challenges related to Keccak-256 are explored. The paper also examines the cross-chain verification, particularly focusing on the interactions between Ethereum and Layer 2 networks. Through careful analysis of storage proof patterns across different network configurations, we identify and formalize three architectures for cross-chain verification. By organizing this complex technical landscape, this analysis provides a structured framework for understanding storage proof implementations in the Ethereum ecosystem, offering insights into their practical applications and limitations.
Blockchain technology has revolutionized decentralized systems by enabling secure, transparent, and tamper-resistant data management. Central to blockchain functionality is the consensus mechanism, which ensures agreement among distributed nodes. This paper presents a comprehensive review of cryptographic solutions underpinning various blockchain consensus protocols. We analyze the role of cryptographic primitives such as hash functions, digital signatures, zero-knowledge proofs, and threshold cryptography in enhancing security and efficiency of consensus algorithms like Proof of Work (PoW), Proof of Stake (PoS), and Practical Byzantine Fault Tolerance (PBFT). The review highlights strengths, limitations, and emerging trends in cryptographic techniques addressing scalability, privacy, and resistance to adversarial attacks. Finally, we discuss future research directions toward integrating advanced cryptography with blockchain consensus to improve trustworthiness and performance.
In recent years, the healthcare sector has encountered significant challenges in authenticating identities for online medical services. A predominant reliance on centralized identity management systems (IDMs) has presented obstacles to the seamless exchange of patient identities among various healthcare institutions, often resulting in data isolation within individual silos. Of paramount concern are the potential privacy breaches associated with centralized IDMs, which may compromise patient confidentiality. In response to these challenges, we propose a novel approach to securely sharing patient details across multiple hospitals utilizing the zero-knowledge access protocol (MediCrypt-ZKAP) within the Hyperledger Fabric blockchain framework. By adopting MediCrypt-ZKAP, hospitals can effectively verify the identities of requesting entities without disclosing sensitive patient information, thereby ensuring the highest levels of confidentiality and privacy protection. The proposed system represents a proactive step towards addressing the critical need for secure and interoperable patient data exchange within the healthcare sector. Through the integration of MediCrypt-ZKAP into existing blockchain infrastructure, our solution aims to enhance data security and privacy while promoting seamless collaboration among healthcare institutions.
Carmit Hazay, Muthuramakrishnan Venkitasubramaniam, Mor Weiss
Abstract Leakage-resilient cryptography aims to protect cryptographic primitives from so-called “side channel attacks” that exploit their physical implementation to learn their input or secret state. Starting from the works of Ishai, Sahai and Wagner (CRYPTO‘03) and Micali and Reyzin (TCC‘04), most works on leakage-resilient cryptography either focus on protecting general computations, such as circuits or multiparty computation protocols, or on specific non-interactive primitives such as storage, encryption, and signatures. This work focuses on leakage resilience for the middle ground, namely for distributed and interactive cryptographic primitives. Our main technical contribution is designing the first secret sharing scheme that is equivocal , resists adaptive probing of a constant fraction of bits from each share, while incurs only a constant blowup in share size. Equivocation is a strong leakage-resilience guarantee, recently introduced by Hazay et al. (ITC, 2021). Our construction is obtained via a general compiler which we introduce, that transforms any secret sharing scheme into an equivocal scheme against adaptive leakage. An attractive feature of our compiler is that it respects additive reconstruction; namely, if the original scheme has additive reconstruction, then the transformed scheme has linear reconstruction. We extend our compiler to a general paradigm for protecting distributed primitives against leakage and show its applicability to various primitives, including secret sharing, verifiable secret sharing, function secret sharing, distributed encryption and signatures, and distributed zero-knowledge proofs. For each of these primitives, our paradigm transforms any construction of the primitive into a scheme that resists adaptive party corruptions, as well as adaptive probing leakage of a constant fraction of bits in each share when the share is stored in memory (but not when it is used in computations). Moreover, the transformation incurs only a constant blowup in the share size and respects additive reconstruction—an important feature for several of these primitives, such as function secret sharing and distributed encryption.
In distributed computing, data trading mechanisms are essential for ensuring the sharing of data across multiple computing nodes. Nevertheless, they currently encounter considerable obstacles, including low accuracy in matching trading parties, ensuring fairness in transactions, and safeguarding data privacy throughout the trading process. In order to address these issues, we put forward a data trading security scheme based on zero-knowledge proofs and smart contracts. In the phase of preparing the security parameters, the objective is to reduce the complexity of generating non-interactive zero-knowledge proofs and to enhance the efficiency of data trading. In the pre-trading phase, we devise attribute atomic matching smart contracts based on precise data property alignment, with the objective of achieving fine-grained matching of data attributes between trading parties. In the trading execution phase, lightweight cryptographic algorithms based on elliptic curve cryptography (ECC) and non-interactive zero-knowledge proofs are employed for the dual encryption of trading data and the generation of attribute proof contracts, thus ensuring the security and privacy of the data. The results of experiments conducted on the Ethereum platform in an industrial IoT scenario demonstrate that our scheme maintains stable and low-cost consumption while ensuring accuracy in matching and privacy protection.
Collision-resistant, cryptographic hash (CRH) functions have long been an integral part of providing security and privacy in modern systems. Certain constructions of zero-knowledge proof (ZKP) protocols aim to utilize CRH functions to perform cryptographic hashing. Standard CRH functions, such as SHA2, are inefficient when employed in the ZKP domain, thus calling for ZK-friendly hashes, which are CRH functions built with ZKP efficiency in mind. The most mature ZK-friendly hash, MiMC, presents a block cipher and hash function with a simple algebraic structure that is well-suited, due to its achieved security and low complexity, for ZKP applications. Although ZK-friendly hashes have improved the performance of ZKP generation in software, the underlying computation of ZKPs, including CRH functions, must be optimized on hardware to enable practical applications. The challenge we address in this work is determining how to efficiently incorporate ZK-friendly hash functions, such as MiMC, into hardware accelerators, thus enabling more practical applications. In this work, we introduce AMAZE, a highly hardware-optimized open-source framework for computing the MiMC block cipher and hash function. Our solution has been primarily directed at resource-constrained edge devices; consequently, we provide several implementations of MiMC with varying power, resource, and latency profiles. Our extensive evaluations show that the AMAZE-powered implementation of MiMC outperforms standard CPU implementations by more than 13$\times$. In all settings, AMAZE enables efficient ZK-friendly hashing on resource-constrained devices. Finally, we highlight AMAZE's underlying open-source arithmetic backend as part of our end-to-end design, thus allowing developers to utilize the AMAZE framework for custom ZKP applications.
The Riemann hypothesis, renowned for its deep connection to the distribution of prime numbers, remains a central problem in mathematics. Understanding the distribution of primes is crucial for developing efficient algorithms and advancing our knowledge of number theory. The Riemann hypothesis is the assertion that all non-trivial zeros are complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. Several equivalent formulations of the Riemann hypothesis exist. Robin's criterion for the Riemann hypothesis is based on an inequality that divisor sum function $\sigma$ must satisfy at natural numbers greater than 5040. We require the properties of superabundant numbers, that is to say left to right maxima of $n \mapsto \frac{\sigma(n)}{n}$. By using Robin's criterion on superabundant numbers, we present a novel approach that culminates in a complete proof of the Riemann hypothesis. This work is an expansion and refinement of the article "Robin's criterion on divisibility", published in The Ramanujan Journal.
The diversity and scarcity of the medical information makes it difficult to create precise global classification approach for the healthcare applications.The main motive is the privacy issue that restricts the data exchanging scope between healthcare institutions.On the contrary, an information from single source is not adequate for developing the worldwide diagnosis approach.The Federated Learning (FL) is a promising solution for privacy and data multiplicity issues, an appropriate aggregation model for multi class and dissimilar medical information is still challenging task in the recognition.Moreover, the FL approaches does not effectively analyzes the each participant execution in the local model and secures the user data.In order to overcome this issue, the Zero-Knowledge Proof (ZKP) based FL approach is developed over blockchain (BC) for performing the COVID-19 classification.The global model of FL uses the two layer Long Short Term Memory (2LLSTM) with federated proximal term (FedProx) namely 2LLSTMFP while the Convolutional Neural Network (CNN) is used in the local model.The integration ZKP and BS is used to improve the data confidentiality while the immutability of BC helps to prevent unauthorized variations for the ledger.The developed FLBC-ZKP is analyzed with two datasets such as COVID-19 Radiography, and CXR images pneumonia and COVID-19.The FLBC-ZKP is evaluated using accuracy, recall, precision, specificity, F1-score, False Negative Rate (FNR) and False Positive Rate (FPR).The existing researches such as WMT, MCCF, 3SFDL and TOTL are used to compare the FLBC-ZKP method.The FLBC-ZKP achieves improved accuracy of 98.34 % for COVID-19 Radiography dataset that is better than the MCCF and 3SFDL.
This research paper explores the CONIKS key management system’s security and efficiency, a system designed to ensure transparency and privacy in cryptographic operations. We conducted a comprehensive analysis of the underlying mathematical principles, focusing on cryptographic hash functions and digital signature schemes, and their implementation in the CONIKS model. Through the use of Merkle trees, we verified the integrity of the system, while zero-knowledge proofs were utilized to ensure the confidentiality of key bindings. We conducted experimental evaluations to measure the performance of cryptographic operations like key generation, signing, and verification with varying key sizes and compared the results against theoretical expectations. Our findings demonstrate that the system performs as predicted by cryptographic theory, with only minor deviations in computational time complexities. The analysis also reveals significant trade-offs between security and efficiency, particularly when larger key sizes are used. These results confirm that the CONIKS system offers a robust framework for secure and efficient key management, highlighting its potential for real-world applications in secure communication systems.
Open access
Cryptography and Data Security
Security in Wireless Sensor Networks
Advanced Steganography and Watermarking Techniques
This paper proposes a new blockchain-based transaction verification infrastructure for co-payment and data verification for multi-modal public transportation systems.Our solution offers a decentralized platform that ensures secure copayments and data integrity while addressing interoperability, data security and transactional transparency.With a private blockchain, transportation providers act as nodes and validated, consensus-approved transactions increase trust and transparency.A standardized data format and robust algorithms for data contribution by transport operators are developed as well as a model for operators, assets, and transactions.Including zero-knowledge proofs improves user privacy by allowing secure authentication without revealing sensitive data.We believe that this research may lead a closer collaboration between public transport operators and provide an enhanced user experience while enabling transport transaction security and data verification.
Blockchain technology has transformed digital transactions by enhancing security and trust through decentralization and cryptography. However, this technology also poses significant privacy challenges in transactions. Specifically, the transparency inherent in blockchain can lead to privacy concerns where sensitive transaction data and user identities are potentially exposed. This paper explores the dichotomy of data privacy and identity privacy within blockchain transactions, using Ethereum as a primary example. It delves into various privacy protection technologies currently in use, such as homomorphic encryption and zero-knowledge proofs, and evaluates their effectiveness and limitations. The paper further discusses the balance between privacy protection and regulatory compliance, which remains a critical challenge. Additionally, it considers future directions for enhancing privacy without compromising the security and functionality of blockchain systems. This exploration is vital for developing blockchain applications that maintain user confidentiality while benefiting from the transparency and security that blockchain provides.
Privacy computing involves the extensive exchange and processing of encrypted data. For the parties involved in these interactions, how to determine the consistency of exchanged data without accessing the original data, ensuring tamper resistance, non-repudiation, quality traceability, indexing, and retrieval during the use of encrypted data, which is a key topic of achieving "Data Availability versus Visibility". This paper proposes a new type of homomorphism: Feature Homomorphism, and based on this feature, introduces a cryptographic scheme for data verification under ciphertext-only conditions. The proposed scheme involves designing a group of algorithms that meet the requirements outlined in this paper, including encryption/decryption algorithms and Feature Homomorphic Algorithm. This group of algorithms not only allows for the encryption and decryption of data but also ensures that the plaintext and its corresponding ciphertext, encrypted using the specified encryption algorithm, satisfy the following property: the eigenvalue of the plaintext obtained using the Feature Homomorphic Algorithm is equal to the eigenvalue of the ciphertext obtained using the same algorithm. With this group of algorithms, it is possible to verify data consistency directly by comparing the eigenvalues of the plaintext and ciphertext without accessing the original data (i.e., under ciphertext-only conditions). This can be used for tamper resistance, non-repudiation, and quality traceability. Additionally, the eigenvalue can serve as a ciphertext index, enabling searchable encryption. This scheme completes a piece of the puzzle in homomorphic encryption. Keywords: Privacy Computing, Data Consistency, Searchable Encryption, Zero-Knowledge Proof, Feature Homomorphism
Michael Olayinka Gbadebo, Ademola Oluwaseun Salako, Oluwatosin Selesi-Aina, Olumide Samuel Ogungbemi · 6 authors
This study examines the effectiveness of current data privacy protocols within cryptocurrency platforms, focusing on encryption strength, anonymity techniques, and AI-powered regulatory compliance tools. Data were sourced from CoinMarketCap and Kaggle, including metrics like Bit Strength, Breach Incidents, and Anonymity Scores, which were analyzed using descriptive statistics, t-tests, and logistic regression. Results showed no significant relationship between encryption strength and breach incidents (p = 0.817), indicating that encryption strength may not be a primary factor in breach prevention. The weak correlation between encryption strength and breaches suggests that other elements, such as platform vulnerabilities or user behaviour, could play a more critical role in security. AI systems, evaluated through metrics like precision (0.168), recall (0.204), and F1 score (0.184), struggled with false positives, showing limitations in accurately detecting breaches and highlighting the need for more refined AI models. Advanced blockchain technologies like Zero-Knowledge Proofs and Homomorphic Encryption enhanced privacy but increased computational costs. It is recommended that hybrid encryption methods be adopted to balance privacy and performance and improve AI systems for more accurate breach detection. Governments must create clear regulations that encourage innovation while ensuring compliance.
In the context of the digital age, data privacy and security issues are increasingly prominent. Blockchain technology plays an important role in data sharing due to its transparency and immutability, but it also brings the risk of privacy leakage. Zero knowledge proof technology provides a solution for verifying data correctness without exposing data content, which is particularly important for blockchain as it can ensure the validity and compliance of transactions while protecting user privacy. Although zero knowledge proof is quite mature in theory, its application in blockchain systems still faces challenges such as computational efficiency, complexity of smart contracts, and system compatibility. This study aims to propose a privacy protection scheme that supports interactive zero knowledge proof by improving the homomorphic encryption Paillier algorithm, in order to enhance the privacy protection capability of blockchain systems and maintain system efficiency and security. The study will adopt an interdisciplinary approach, combining cryptography, computer science, and network security theory, to deeply analyze the application effect of zero knowledge proof technology in blockchain, explore its optimization space and applicability.
A. Jabbari, Gowri Ramachandran, Sidra Malik, Raja Jurdak
In the current digital landscape, supply chains have transformed into complex networks driven by the Internet of Things (IoT), necessitating enhanced data sharing and processing capabilities to ensure traceability and transparency. Leveraging Blockchain technology in IoT applications advances reliability and transparency in near-real-time insight extraction processes. However, it raises significant concerns regarding data privacy. Existing privacy-preserving approaches often rely on Smart Contracts for automation and Zero Knowledge Proofs (ZKP) for privacy. However, apart from being inflexible in adopting system changes while effectively protecting data confidentiality, these approaches introduce significant computational expenses and overheads that make them impractical for dynamic supply chain environments. To address these challenges, we propose ZK-DPPS, a framework that ensures zero-knowledge communications without the need for traditional ZKPs. In ZK-DPPS, privacy is preserved through a combination of Fully Homomorphic Encryption (FHE) for computations and Secure Multi-Party Computations (SMPC) for key reconstruction. To ensure that the raw data remains private throughout the entire process, we use FHE to execute computations directly on encrypted data. The "zero-knowledge" aspect of ZK-DPPS refers to the system's ability to process and share data insights without exposing sensitive information, thus offering a practical and efficient alternative to ZKP-based methods. We demonstrate the efficacy of ZK-DPPS through a simulated supply chain scenario, showcasing its ability to tackle the dual challenges of privacy preservation and computational trust in decentralised environments.
Enterprise resource planning (ERP) platforms integrate critical business operations, including manufacturing, inventory, accounting, human resources, and supply chain management, into unified systems for automation and analytics. However, traditional centralized ERP architectures have notable limitations around security, transparency, costs, and process integrity. ERP data silos inhibit trust, provenance tracking, and collaboration across organizational boundaries. Recently, blockchain distributed ledger technology has emerged as a promising approach to transforming ERP by enabling decentralized verifiable workflows, immutable record keeping, and end-to-end transaction traceability. Blockchain offers a paradigm shift for ERP by distributing control across a peer-to-peer network organized around consensus, cryptography, and innovative algorithms. Early research and prototypes demonstrate blockchain's potential to enhance security, trust, provenance, automation, and standardization in enterprise systems. Blockchain ERP pilots show feasibility in supply chain tracking, accounting, procurement, manufacturing, and more. By sharing tamper-evident ledgers across companies, blockchain builds transparency, integrity, and collaboration into ERP workflows. Several technical concepts underpin blockchain ERP capabilities. Distributed ledgers cryptographically chain transaction records. Consensus protocols like proof-of-work and Byzantine fault tolerance enable unanimous agreement on valid state changes across nodes. Smart contracts automate multi-step workflows based on predefined conditions. Hashing, public-key encryption, and zero-knowledge proofs provide security and privacy. Together, these constructs allow decentralized and verifiable ERP processes. However, challenges remain prior to enterprise adoption at scale. Blockchain ERP must still overcome hurdles integrating with legacy systems, coordinating complex cross-organizational ecosystems, scaling transaction throughput and data storage, and reducing implementation costs. Data privacy also requires consideration under public blockchain models. In conclusion, blockchain shows immense yet nascent promise for revolutionizing outdated ERP platforms once integration, coordination, scaling, and cost obstacles are surmounted through further research and development. By transforming how enterprises architect and optimize mission-critical systems, blockchain may profoundly disrupt enterprise computing and business processes. The technology remains in its early stages but holds revolutionary potential for ERP and inter-organizational collaboration.
Redmond R. Shamshiri, Abdullah Kaviani Rad, Maryam Behjati, Siva K. Balasundram
The challenges and drawbacks of manual weeding and herbicide usage, such as inefficiency, high costs, time-consuming tasks, and environmental pollution, have led to a shift in the agricultural industry toward digital agriculture. The utilization of advanced robotic technologies in the process of weeding serves as prominent and symbolic proof of innovations under the umbrella of digital agriculture. Typically, robotic weeding consists of three primary phases: sensing, thinking, and acting. Among these stages, sensing has considerable significance, which has resulted in the development of sophisticated sensing technology. The present study specifically examines a variety of image-based sensing systems, such as RGB, NIR, spectral, and thermal cameras. Furthermore, it discusses non-imaging systems, including lasers, seed mapping, LIDAR, ToF, and ultrasonic systems. Regarding the benefits, we can highlight the reduced expenses and zero water and soil pollution. As for the obstacles, we can point out the significant initial investment, limited precision, unfavorable environmental circumstances, as well as the scarcity of professionals and subject knowledge. This study intends to address the advantages and challenges associated with each of these sensing technologies. Moreover, the technical remarks and solutions explored in this investigation provide a straightforward framework for future studies by both scholars and administrators in the context of robotic weeding.
This review examines how blockchain technology can be leveraged to enhance data privacy and security in sustainable supply chain management (SSCM). As global supply chains become increasingly complex and the demand for sustainability grows, ensuring data privacy and security has become a critical concern. Traditional supply chain systems often face challenges such as data breaches, lack of transparency, and difficulty in tracing products and materials. Blockchain technology, with its decentralized, immutable, and transparent architecture, offers a promising solution to these challenges. Blockchain can enhance data security by ensuring that data is tamper-proof, traceable, and encrypted, thus protecting sensitive information across the supply chain. It provides transparency while allowing permissioned access, ensuring that stakeholders can verify data without exposing confidential information. Furthermore, privacy-preserving technologies such as zero-knowledge proofs and homomorphic encryption allow verification of data without compromising its security. Smart contracts enable automated compliance with regulatory frameworks like GDPR, reducing the risk of human error and improving operational efficiency. The integration of blockchain in SSCM can improve traceability, transparency, and accountability, thereby promoting environmental and social sustainability. By tracking the origin and journey of goods, blockchain helps verify ethical sourcing practices and reduce carbon footprints. However, the technology also presents challenges, including scalability, integration with legacy systems, and cost considerations. Through case studies in industries such as food, textiles, and renewable energy, this review highlights the practical applications and benefits of blockchain for SSCM. It concludes that blockchain has the potential to revolutionize supply chain operations, but careful consideration must be given to overcoming its technical and financial barriers to widespread adoption.
The world is rapidly heading towards digitization and digital documentation.The COVID-19 pandemic has highlighted the significance of digitization in our daily lives.Nowadays, fake documents are widely available and easy to obtain, harming both our financial system and social trust.Consequently, there is a growing demand for procedures to verify and authenticate various crucial documents, including transactional, financial, governmental, and personal certificates, as well as educational certificates.This type of practice can be done using blockchain and cryptography technology.In this paper, we propose a Blockchain based Digital Identity Management System (BDIMS) that empowers organizations to generate instantaneously authenticated and tamper-resistant digital credentials.It issues a signed document and stores the signature on the blockchain.The verifier can easily verify the signature from the blockchain instantly using the digital signature concept.BDIMS also provides a QR-code system for real-time identity verification.It also introduces zero-knowledge proof for verifying the part of an identity without revealing the original statement.Furthermore, a user can store and share all their identities on a single platform using BDIMS.The proposed model effectively addresses the shortcomings of traditional methods by ensuring a comprehensive and streamlined approach.It successfully bridges the gaps and overcomes the difficulties inherent in conventional document verification systems, meeting all the necessary criteria for a robust and reliable verification process.
Zero-knowledge layer 2 protocols emerge as a compelling approach to overcoming blockchain scalability issues by processing transactions through the transaction finalization process. During this process, transactions are efficiently processed off the main chain. Besides, both the transaction data and the zero-knowledge proofs of transaction executions are reserved on the main chain, ensuring the availability of transaction data as well as the correctness and verifiability of transaction executions. Hence, any bugs that cause the transaction finalization failure are crucial, as they impair the usability of these protocols and the scalability of blockchains. In this work, we conduct the first systematic study on finalization failure bugs in zero-knowledge layer 2 protocols, and define two kinds of such bugs. Besides, we design fAmulet, the first tool to detect finalization failure bugs in Polygon zkRollup, a prominent zero-knowledge layer 2 protocol, by leveraging fuzzing testing. To trigger finalization failure bugs effectively, we introduce a finalization behavior model to guide our transaction fuzzer to generate and mutate transactions for inducing diverse behaviors across each component (e.g., Sequencer) in the finalization process. Moreover, we define bug oracles according to the distinct bug definitions to accurately detect bugs. Through our evaluation, fAmulet can uncover twelve zero-day finalization failure bugs in Polygon zkRollup, and cover at least 20.8% more branches than baselines. Furthermore, through our preliminary study, fAmulet uncovers a zero-day finalization failure bug in Scroll zkRollup, highlighting the generality of fAmulet to be applied to other zero-knowledge layer 2 protocols. At the time of writing, all our uncovered bugs have been confirmed and fixed by Polygon zkRollup and Scroll zkRollup teams.