In blockchain ecosystems, maintaining transparency and privacy has become an ethical dilemma. This is because, while certain specific information of the user is shared to ensure transparency of transactions across networks, such information could be detrimental to the user, as there is a possibility of it being tampered with. For instance, in the Catalyst voting process in Cardano, users can still see the amount of ADA tokens being held by other users, which can influence their voting options, especially when large ADA holders vote in support of certain ideas or proposals. To discourage such challenges as voter manipulation and vote buying, this study proposed the implementation of zero-knowledge proof (ZKP) in blockchain ecosystems to enhance the transparency of the catalyst voting process and enhance efficiency and speed of result release. Using survey questionnaire and a multivocal literature review, this study was able to proof that ZKP cannot only be applied in the catalyst voting process to enhance its transparency, but also addressed potential challenges to its applications such as scalability, encourage trust and fairness of the voting system, and improve voter participation due to its user-friendliness. Mathematical models emphasize scaled voting as optimal for balancing inclusion and plutocratic control.
To address the issues of excessive centralization and difficulties in cross-domain identity authentication in traditional cross-domain authentication schemes, an anonymous cross-domain authentication framework based on blockchain smart contracts is proposed. The framework uses Pedersen commitments to construct non-interactive zero-knowledge proofs, enabling users to achieve anonymous identity authentication with low communication overhead. Blockchain smart contracts are employed to record information about users and domains, realizing decentralized information storage. Experimental results show that the proposed scheme effectively reduces the level of centralization in cross-domain authentication and improves both the efficiency and security of cross-domain identity authentication.
Cryptography and Data Security
Advanced Steganography and Watermarking Techniques
In the era of decentralized identity management within blockchain ecosystems, ensuring user privacy during authentication processes is a critical concern. This dissertation addresses the challenge of privacypreserving authentication within decentralized identity management systems, specifically on the Algorand blockchain platform. As digital identity solutions become critical in todayâs interconnected world, traditional centralized models expose user data to substantial privacy and security risks, such as data breaches, identity theft, and unauthorized access. The research leverages Algorandâs pure proof of stake (PPoS) consensus mechanism, recognized for its scalability and energy efficiency, along with cryptographic techniques such as zero-knowledge proofs (ZKPs) and the Pedersen commitment scheme. The primary contribution of this dissertation is the development of a proof of concept decentralized application (DApp) designed for secure and anonymous voting, designed to balance data protection with usability within the context of Dharma Teams, a decentralized application of Yari Labs. By incorporating cryptographic primitives such as anonymous credentials and secure, decentralized authentication protocols, the DApp demonstrates how user privacy can be maintained even in open blockchain environments. The framework developed within this research not only ensures user anonymity, but also upholds the integrity and transparency of the authentication process. Furthermore, this dissertation explores the applicability of these privacy-preserving methods in various use cases, including decentralized finance, supply chain management, and verification of digital identity. Through a blend of theoretical insights and practical implementation, this work lays a pathway toward more secure and use centric digital ecosystems on Algorand and similar platforms.
Kelvin, Richard Raymond, Fahrur Rozzi Nasution, Andrew Law · 7 authors
Teknologi blockchain, dengan karakteristiknya yang desentralisasi, transparan, dan tahan tamper, menawarkan solusi inovatif untuk mengatasi tantangan dalam sistem pembayaran tradisional. Dengan mekanisme konsensus seperti Proof of Work (PoW) dan Proof of Stake (PoS), blockchain dapat menjamin integritas data dan mencegah pengeluaran ganda. Keamanan transaksi terjaga melalui penggunaan kriptografi canggih, yang juga melindungi identitas pengguna. Meski demikian, implementasi blockchain menghadapi tantangan signifikan, termasuk skalabilitas, konsumsi energi, serta kompleksitas regulasi. Studi ini menganalisis pengaruh blockchain terhadap keamanan dan privasi dalam sistem pembayaran berbasis kriptografi, termasuk evaluasi teknik seperti Zero-Knowledge Proofs dan enkripsi homomorfik. Melalui pendekatan kualitatif berbasis analisis literatur dan studi kasus, penelitian ini mengeksplorasi aplikasi blockchain dalam berbagai sektor pembayaran, seperti pembayaran lintas batas dan micropayments, serta dampaknya terhadap peran lembaga keuangan. Hasilnya menunjukkan bahwa blockchain berpotensi meningkatkan efisiensi dan keamanan transaksi, meskipun isu regulasi dan privasi masih menjadi kendala utama untuk adopsi lebih luas.
Autonomous AI agents present transformative opportunities and significant governance challenges. Existing frameworks, such as the EU AI Act and the NIST AI Risk Management Framework, fall short of addressing the complexities of these agents, which are capable of independent decision-making, learning, and adaptation. To bridge these gaps, we propose the ETHOS (Ethical Technology and Holistic Oversight System) framework, a decentralized governance (DeGov) model leveraging Web3 technologies, including blockchain, smart contracts, and decentralized autonomous organizations (DAOs). ETHOS establishes a global registry for AI agents, enabling dynamic risk classification, proportional oversight, and automated compliance monitoring through tools like soulbound tokens and zero-knowledge proofs. Furthermore, the framework incorporates decentralized justice systems for transparent dispute resolution and introduces AI specific legal entities to manage limited liability, supported by mandatory insurance to ensure financial accountability and incentivize ethical design. By integrating philosophical principles of rationality, ethical grounding, and goal alignment, ETHOS aims to create a robust research agenda for promoting trust, transparency, and participatory governance. This innovative framework offers a scalable and inclusive strategy for regulating AI agents, balancing innovation with ethical responsibility to meet the demands of an AI-driven future.
Rafiqul Islam Munna, Kazi Md. Rokibul Alam, Yasuhiko Morimoto
Data sharing across collaborative mining can aid the community through analyses for decision-making tasks. While sharing personal data with a 3rdparty, data anonymization is a lawful obligation. Besides, preventing unauthorized access is another demand for storing data in a database. This paper proposes a framework of location data anonymization and data privacy over the database to ensure utility while data mining. For anonymization, upon the location data, it exploits dynamic geofencing, gridding, differential privacy, and zero-knowledge proof consecutively to perform the geographical analyses. Also, to securely store the data in the database, it encrypts data by elliptic curve cryptography. The anonymization framework produces grid-based circular coordinate boundaries, adds controlled noise, and proves the 3rdparty about its query results without telling any knowledge of the location (i.e., coordinates). Finally, the performance evaluation, analyses, comparisons, etc., using real data demonstrate that the proposed framework retains a better trade-off between the data utility and anonymity than the state-of-the-art works.
Mihaela Ifrim, Ben Pineau, Daniel Tataru, Mitchell A. Taylor
In this article, we provide a definitive well-posedness theory for the free boundary problem in incompressible magnetohyrodynamics. Despite the clear physical interest in this system and the remarkable progress in the study of the free boundary Euler equations in recent decades, the low regularity well-posedness of the free boundary MHD equations has remained completely open. This is due, in large part, to the highly nonlinear wave-type coupling between the velocity, magnetic field and free boundary, which has forced previous works to impose restrictive geometric constraints on the data. To address this problem, we introduce a novel Eulerian approach and an entirely new functional setting, which better captures the wave equation structure of the MHD equations and permits a complete Hadamard well-posedness theory in low-regularity Sobolev spaces. In particular, we give the first proofs of existence, uniqueness and continuous dependence on the data at the sharp $s>\frac{d}{2}+1$ Sobolev regularity, in addition to a blowup criterion for smooth solutions at the same low regularity scale. Moreover, we provide a completely new method for constructing smooth solutions which, to our knowledge, gives the first proof of existence (at any regularity) in our new functional setting. All of our results hold in arbitrary dimensions and in general, not necessarily simply connected, domains. By taking the magnetic field to be zero, they also recover the corresponding sharp well-posedness theorems for the free boundary Euler equations. The methodology and tools that we employ here can likely be fruitfully implemented in other free boundary models.
This study focuses on data encryption and privacy protection technologies in cloud computing environments. By systematically implementing and evaluating various encryption algorithms (such as AES, RSA, and homomorphic encryption) and privacy protection techniques (including data masking, differential privacy, secure multi-party computation, and zero-knowledge proofs), the feasibility and effectiveness of these technologies in cloud environments are explored. A simulated cloud environment was constructed for experiments, and the results indicate that AES performs excellently in large-scale data processing, while homomorphic encryption demonstrates unique advantages in specific scenarios. Privacy protection techniques can achieve a balance between protecting user privacy and maintaining data availability. System performance and security tests confirm that the proposed solutions effectively support the data security and privacy protection needs in large-scale cloud environments. This research provides a comprehensive technical implementation and evaluation reference for data security and privacy protection in cloud computing environments, while also highlighting some challenges and offering valuable insights for future research directions.
The advancement of 6G wireless communication technology has facilitated the integration of Vehicular Ad-hoc Networks (VANETs). However, the messages transmitted over the public channel in the open and dynamic VANETs are vulnerable to malicious attacks. Although numerous researchers have proposed authentication schemes to enhance the security of Vehicle-to-Vehicle (V2V) communication, most existing methodologies face two significant challenges: (1) the majority of the schemes are not lightweight enough to support real-time message interaction among vehicles; (2) the sensitive information like identity and position is at risk of being compromised. To tackle these issues, we propose a lightweight dual authentication protocol for V2V communication based on Physical Unclonable Function (PUF). The proposed scheme accomplishes dual authentication between vehicles by the combination of Zero-Knowledge Proof (ZKP) and MASK function. The security analysis proves that our scheme provides both anonymous authentication and information unlinkability. Additionally, the performance analysis demonstrates that the computation overhead of our scheme is approximately reduced 23.4% compared to the state-of-the-art schemes. The practical simulation conducted in a 6G network environment demonstrates the feasibility of 6G-based VANETs and their potential for future advancements.
Chunyu Lu, Jun Luo, Duo Shang, Haoran Li · 8 authors
This paper proposes a novel cross-domain access control mechanism for industrial Internet alliance chains using zero-knowledge proof (ZKP). The mechanism integrates ZKP with blockchain technology to achieve secure, privacy-preserving, and decentralized access control across multiple industrial domains. A formal system model and security framework are presented, along with an efficient ZKP protocol for attribute-based access control. The proposed scheme is implemented using smart contracts and distributed ledger technology. Comprehensive performance analysis and security verification demonstrate the mechanismâs high throughput, low latency, and resilience against various security threats. Experimental results show that the proposed solution is scalable and practical for large-scale industrial Internet scenarios, enabling secure data sharing and collaboration while protecting user privacy and ensuring fine-grained access control.
In order to enable more types of machine learning models to use zero-knowledge proofs to enhance their computational verifiability, this study proposes a zero-knowledge machine learning conversion method based on the Taylor series. Firstly, a polynomial expansion of structures with transcendental functions in ordinary machine learning models is performed using Taylor's formula. The corresponding arithmetic circuit descriptions are written in ZKP based on the converted model structures. Finally, the proof body is generated, which allows the verifier to verify the correctness of the results quickly. The basic experimental idea is also given, and the scheme's feasibility is verified, which can be done to provide a verification path for the model without seriously affecting its accuracy.
Frequent user data breaches and misuse incidents highlight the flaws in current identity management systems. This study proposes a blockchain-based, peer-supervised self-sovereign identity (SSI) generation and privacy protection technology. Our approach creates unique digital identities on the blockchain, enabling secure cross-domain recognition and data sharing and satisfying the essential users' requirements for SSI. Compared to existing SSI solutions, our approach has the practical advantages of less implementation cost, ease of users' understanding and agreement, and better possibility of being soon adopted by current society and legal systems. The key innovative technical features include (1) using a zero-knowledge proof technology to ensure data remain "usable but invisible", mitigating data breach risks; (2) introducing a peer review mechanism among service providers to prevent excessive data requests and misuse; and (3) implementing a comprehensive multi-party supervision system to audit all involved parties and prevent misconduct.
The current digital landscape is constantly under cybersecurity risks and data breaches, making trust and security systems of paramount concern. Blockchain technology offers promising solutions to enhance security and transparency in various sectors. However, it is important to understand the risks associated and proceed with caution with this revolutionary technology, as countless individuals and organizations may be left vulnerable to the fear of data breaches and regulatory pitfalls. This fear, therefore, should not be taken lightly. The struggle to ensure this technology aligns with security compliance regulations is not just a technological challenge- it is integral to safeguarding personal identity, businesses, and our futures in an increasingly uncertain world. As such, this paper looks to examine the intricate relationship between blockchain technology and security compliance, exploring the challenges, solutions, and future implications of this Blockchain technology. Keywordsâ Blockchain technology, security compliance, data privacy, GDPR, selective mutability, zero-knowledge proofs, smart contracts, auditing, industry-specific approaches, healthcare, supply chain management, AML, KYC
The acceleration of the digital transformation process imposed by the pandemic in all the countries of the European Union, and in all sectors, has given way to a revolution that up until a couple of years ago would have been impossible even to imagine. Digital innovation has become a factor of competitiveness in all sectors. In this new scenario that has come to be, the Blockchain technology, the Self-Sovereign Identity paradigm, Internet of Things, and, in general, the new technologies that will emerge, will constitute enhancers of competitiveness and will have to aim for interoperability. In this context, this article develops and presents a model proposal in the healthcare field that aims to highlight how the combination of the Blockchain technology and the Self-Sovereign Identity paradigm restores full control over a personâs identity and information, while ensuring the integrity of all medical reports, enabling secure communications between personal medical devices and patient/doctor applications on devices exploiting peer Decentralized Identifiers and ensuring data privacy, exploiting Zero Knowledge Proofs. The proposal relies on the Veramo platform, treating all medical reports as verifiable credentials and storing them in digital wallets owned by the patient. The article concludes by presenting a prototype designed and implemented for managing medication prescriptions, their issuance, and their exchange.
With the widespread adoption and increasing application of blockchain technology, cryptocurrency wallets used in Bitcoin and Ethereum play a crucial role in facilitating decentralized asset management and secure transactions. However, wallet security relies heavily on private keys, with insufficient attention to the risks of theft and exposure. To address this issue, Chaum et al. (ACNSâ21) proposed a âproof of ownershipâ method using a âbackup keyâ to prove ownership of private keys even when exposed. However, their interactive proof approach is inefficient in large-scale systems and vulnerable to side-channel attacks due to the long key generation time. Other related schemes also suffer from low efficiency and complex key management, increasing the difficulty of securely storing backup keys. In this paper, we present an efficient, non-interactive proof generation approach for ownership of secret keys using a single backup key. Our approach leverages non-interactive zero-knowledge proofs and symmetric encryption, allowing users to generate multiple proofs with one fixed backup key, simplifying key management. Additionally, our scheme resists quantum attacks and provides a fallback signature. Our new scheme can be proved to capture unforgeability under the computational indistinguishability from the Uniformly Random Distribution property of a proper hash function and soundness in the quantum random oracle model. Experimental results indicate that our approach achieves a short key generation time and enables an efficient proof generation scheme in large-scale decentralized systems. Compared with state-of-the-art schemes, our approach is applicable to a broader range of scenarios due to its non-interactive nature, short key generation time, high efficiency, and simplified key management system.
Katharina Koschatko, Reinhard LĂŒftenegger, Christian Rechberger
Gröbner basis cryptanalysis of hash functions and ciphers, and their underlying permutations, has seen renewed interest recently. Anemoi (Cryptoâ23) is a permutation-based hash function that is efficient for a variety of arithmetizations used in zero-knowledge proofs. In this paper, exploring both theoretical bounds as well as experimental validation, we present new complexity estimates for Gröbner basis attacks on the Anemoi permutation over prime fields.We cast our findings in what we call the six worlds of Gröbner basis cryptanalysis. As an example, keeping the same security arguments of the design, we conclude that at least 41 instead of 37 rounds would need to be used for 256-bit security, whereby our suggestion does not yet include a security margin.
Open access
Polynomial and algebraic computation
Cryptography and Residue Arithmetic
Mathematics, Computing, and Information Processing
Traditional identity management systems, often centralized, face challenges around privacy, data security, and user control, leaving users vulnerable to data breaches and misuse. This paper explores the potential of using the Arweave network to develop an identity management solution. By harnessing Arweave's permanent storage, our solution offers the users a Self-Sovereign Identity (SSI) framework, that uses Decentralized Identifiers (DIDs) and Verifiable Credentials (VCs) to allow individuals and other entities to create, own, and manage their digital identities. Further, the solution integrates privacy-preserving technologies, including zero-knowledge proofs and the BBS(+) signature scheme, enabling selective disclosure. This approach ultimately enhances user privacy and supports compliance with European Union legislation and regulatory standards like the General Data Protection Regulation (GDPR) by design.
Yalan Wang, Liqun Chen, Long Meng, Christopher J. P. Newton
Concerns about how third parties manage personal information have led to the development of decentralized identities (DIDs) and verifiable credentials (VCs). The World Wide Web Consortium (W3C) working group has been developing standards for DIDs and VCs. In the W3C standards, a DID identifies an entity (a DID holder) and a VC confirms that this DID holder has some associated attributes. A DID holder can obtain many VCs and confirm any number of these VCs to others (verifiers) in verifiable presentations (VPs). In order to keep a holderâs identity and attributes private, it is necessary to achieve anonymous VPs that allows this information to be kept confidential. The W3C working group recommends using randomizable signatures to create VCs with zero-knowledge proofs for this purpose. However, the anonymous VPs provided by the this method are limited that in the real world, credentials in cross domains cannot be universally verified. To overcome this limitation, in this paper, we propose a new scheme, called Verifiable Credentials with anonymous DIDs (VCaDID), which aims to achieve anonymous VPs in cross-domain settings. The main technique in our VCaDID scheme is a ring signature with multiple attributes by hiding a holderâs public key among a ring of holders. In our scheme, we set private keys associated with the holderâs DID and attributes, which allow the holder to anonymously present these credentials in a verifiable way. We also prove that the proposed VCaDID scheme satisfies correctness, anonymity and unforgeability under security assumptions of discrete log and random oracle model. Finally, we implement our scheme to demonstrate its feasibility.
Due to security risks such as data theft, data tampering, and replay attacks during data sharing, data owners need to establish a trust relationship before sharing their data. To reduce the burdens of data retrieval, how to generate an efficient and secure index has become a challenge. Crypto-based solutions currently only encrypt the data to ensure confidentiality, which ignore the integrity and correctness of the sharing data. An hash-based authentication structures called Merkle Tree is an effective tool for data integrity verification. However, it incurs high maintenance costs when dealing with dynamically changing datasets and is limited by data modalities. In this paper, we propose a trustworthy index for multimodal data sharing (MT-Index). Specifically, we propose a Merkle tree based on the Semantic Web, replacing traditional data labels with semantic graphs to represent the content of data blocks. We utilize timestamp to mark the update operations and store them in the leaf nodes of the tree, so as to effectively handle multimodal data and dynamic dataset changes. Additionally, we design a trust network based on zero-knowledge proofs, utilizing ZK-STARKs to ensure the credibility of one-to-one interactions, and implement a two-round protocol to verify the trust mechanism among multiple parties. Through formal analysis, we demonstrate that MT-Index achieves the desired security objectives with minimal storage and generation overhead.
With the rapid advancement and application of the Internet of Medical Things (IoMT), personal health records (PHRs) are now increasingly comprised of data collected by Internet of Things (IoT) devices and medical records documented by healthcare professionals. Personal health record (PHR) sharing demonstrates great potential in improving the accuracy of disease diagnosis. However, PHR sharing also brings risks such as illegal access and personal information leakage. Some works explored using blockchain or attribute-based encryption (ABE) to solve these privacy leakage problems, but those solutions did not pay attention to the userâs attribute privacy. In this work, we combine a linear secret sharing scheme (LSSS) and zero-knowledge succinct non-interactive argument of knowledge (zkSNARK) scheme to design an efficient zero-knowledge proof protocol called zk-AHSNARK. It can verify the userâs attribute permissions while also hiding attribute information. Based on zk-AHSNARK, we propose a novel PHR sharing scheme that protects attribute privacy. Data security is ensured by storing encrypted data in the interplanetary file system (IPFS). In addition, we introduce keyword ciphertext search to achieve fast data retrieval, and we implement the search and verification algorithms via a smart contract, ensuring the trustworthiness and integrity of the execution. Finally, through a large number of simulations, we demonstrated the suggested schemeâs viability and security.
Decentralized applications (DApps) are increasingly using off-chain data, yet growing concerns about data privacy hinder their widespread adoption. Zero-knowledge proofs (ZKPs) have emerged as a solution to this problem. This paper proposes a novel hybrid framework that combines Zero-Knowledge Scalable Transparent Arguments of Knowledge (zk-STARKs) and Zero-Knowledge Succinct Non-Interactive Arguments of Knowledge (zk-SNARKs) to deliver an efficient, scalable, quantum-resistant, and privacy-preserving ZKP system. The framework employs zk-STARKs' ability to handle large computations with quantum resistance and zk-SNARKs' succinct proofs and fast verification. The hybrid framework utilizes advanced arithmetization techniques to balance scalability, privacy, and security, including Algebraic Intermediate Representation (AIR) for zk-STARKs and Rank-1 Constraint Systems (RlCS) or PLONKish constraints for zk-SNARKs. The framework uses the Kate-Zaverucha-Goldberg (KZG) polynomial commitment scheme for reliability and transparency and eliminates trusted setup by using Discrete-logarithm-based Argument of Recursive Knowledge (DARK) commitments and Poseidon hashing. This paper details the construction of the hybrid proof system, analyzes its complexity, and explores its potential applications. The framework solves smart contract data feed limitations, security, efficiency, and scalability while prioritizing privacy.
Ze-Peng Yang, Wenlong Tian, Emma Zhang, Zhiyong Xu
With the widespread adoption of cloud storage, effectively identifying and eliminating redundant data among users while ensuring data security has become a significant challenge. However, traditional similarity detection methods has limitations in privacy protection. Although conventional encryption techniques can safeguard privacy, they have difficulty detecting redundancy between similar blocks. Thus, we propose a secure reduction of redundant and similar data for cloud storage to address these challenges based on zero-knowledge proof (Sec-Reduce), called Sec-Reduce. It first employs a novel zero-knowledge proof technique for file-level redundancy detection, where redundant files are identified and excluded from storage. To further determine the similarity of non-redundant files, the scheme performs content-based chunking and feature extraction using a similarity feature extraction method. These extracted features are then encrypted using the approximate homomorphic encryption scheme Cheon-Kim-Kim-Song (CKKS) to enable similarity detection in the ciphertext environment. Finally, secure delta encoding is applied to store unique ciphertext blocks and deltas. Evaluations of real-world datasets demonstrate that Sec-Reduce achieves higher storage savings than existing encrypted storage methods, with storage overhead comparable to plaintext storage and only moderate performance overhead.
Jianwei Liu, Wei Wan, Chun Long, Jing Li · 6 authors
In zero-trust environments, fair exchange schemes have long faced challenges of low efficiency and high computational overhead when verifying the integrity of large-scale data. To address these issues, this paper proposes ZKFDT, an efficient data fair exchange scheme based on optimized zero-knowledge proof algorithms, offering improvements in efficiency, fairness, and security. In terms of efficiency, ZKFDT leverages IPFSâs hash-based addressing mechanism to significantly reduce network communication overhead compared to traditional data transmission methods, while also optimizing the multi-scalar multiplication algorithm, improving proof generation efficiency by 2x. Regarding security, ZKFDT adopts the more secure ABR23 protocol, addressing the malleability attack vulnerabilities of Groth16 while maintaining its low communication overhead. Through the implementation of smart contracts, including proof verification, atomic swaps, and time-lock functionality, ZKFDT ensures fairness and immutability in data transactions. Experimental results show that ZKFDT demonstrates high efficiency and practical feasibility in large-scale data transaction applications.
Shahzad Ahmad Butt, Benjamin Reynolds, V. Ramamurthy, Xiao Xiao · 8 authors
Zero-Knowledge Proofs (ZKPs) have emerged as an important cryptographic technique allowing one party (prover) to prove the correctness of a statement to some other party (verifier) and nothing else. ZKPs give rise to user's privacy in many applications such as blockchains, digital voting, and machine learning. Traditionally, ZKPs suffered from poor scalability but recently, a sub-class of ZKPs known as Zero-knowledge Succinct Non-interactive ARgument of Knowledges (zk-SNARKs) have addressed this challenge. They are getting significant attention and are being implemented by many public libraries. In this paper, we present a novel scalable architecture that is suitable for accelerating the zk-SNARK prover compute on FPGAs. We focus on the multi-scalar multiplication (MSM) that accounts for the majority of computation time spent in zk-SNARK systems. The MSM calculations extensive rely on modular arithmetic so highly optimized Intel IP Libraries for modular arithmetic are used. The proposed architecture exploits the parallelism inherent to MSM and is implemented using the Intel OneAPI framework for FPGAs. Our implementation runs 110x-150x faster compared to reference software library, uses a generic curve form in Jacobian coordinates and is the first to report FPGA hardware acceleration results for BLS12-381 and BN128 family of elliptic curves.