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Zero-knowledge proofs have emerged as a powerful tool for enhancing privacy and security in blockchain applications. However, the efficiency and scalability of proof systems remain a significant challenge, particularly in the context of Merkle tree inclusion proofs. Traditional proof aggregation techniques based on AND logic suffer from high verification complexity and data communication overhead, limiting their practicality for large-scale applications. In this paper, we propose a novel proof aggregation approach based on OR logic, which enables the generation of compact and universally verifiable proofs for Merkle tree inclusion. By aggregating proofs using OR logic, we achieve a proof size that is independent of the number of leaves in the tree, and verification can be performed using any single valid leaf hash. This represents a significant improvement over AND aggregation, which requires the verifier to process all leaf hashes. We formally define the OR aggregation logic, describe the process of generating universal proofs, and provide a comparative analysis demonstrating the advantages of our approach in terms of proof size, verification data, and universality. Furthermore, we discuss the potential of combining OR and AND aggregation logics to create complex acceptance functions, enabling the development of expressive and efficient proof systems for various blockchain applications. The proposed techniques have the potential to significantly enhance the scalability, efficiency, and flexibility of zero-knowledge proof systems, paving the way for more practical and adaptive solutions in the blockchain ecosystem.

Addressing the critical challenge of ensuring data integrity in decentralized systems, this paper delves into the underexplored area of data falsification probabilities within Merkle Trees, which are pivotal in blockchain and Internet of Things (IoT) technologies. Despite their widespread use, a comprehensive understanding of the probabilistic aspects of data security in these structures remains a gap in current research. Our study aims to bridge this gap by developing a theoretical framework to calculate the probability of data falsification, taking into account various scenarios based on the length of the Merkle path and hash length. The research progresses from the derivation of an exact formula for falsification probability to an approximation suitable for cases with significantly large hash lengths. Empirical experiments validate the theoretical models, exploring simulations with diverse hash lengths and Merkle path lengths. The findings reveal a decrease in falsification probability with increasing hash length and an inverse relationship with longer Merkle paths. A numerical analysis quantifies the discrepancy between exact and approximate probabilities, underscoring the conditions for the effective application of the approximation. This work offers crucial insights into optimizing Merkle Tree structures for bolstering security in blockchain and IoT systems, achieving a balance between computational efficiency and data integrity., Comment: 7 pages

The scalability of blockchain technology remains a pivotal challenge, impeding its widespread adoption across various sectors. This study introduces an innovative approach to address this challenge by proposing the adaptive restructuring of Merkle and Verkle trees, fundamental components of blockchain architecture responsible for ensuring data integrity and facilitating efficient verification processes. Unlike traditional static tree structures, our adaptive model dynamically adjusts the configuration of these trees based on usage patterns, significantly reducing the average path length required for verification and, consequently, the computational overhead associated with these processes. Through a comprehensive conceptual framework, we delineate the methodology for adaptive restructuring, encompassing both binary and non-binary tree configurations. This framework is validated through a series of detailed examples, demonstrating the practical feasibility and the efficiency gains achievable with our approach. Moreover, we present a comparative analysis with existing scalability solutions, highlighting the unique advantages of adaptive restructuring in terms of simplicity, security, and efficiency enhancement without introducing additional complexities or dependencies. This study's implications extend beyond theoretical advancements, offering a scalable, secure, and efficient method for blockchain data verification that could facilitate broader adoption of blockchain technology in finance, supply chain management, and beyond. As the blockchain ecosystem continues to evolve, the principles and methodologies outlined herein are poised to contribute significantly to its growth and maturity.

In the rapidly evolving landscape of blockchain technology, ensuring the integrity and security of data is paramount. This study delves into the security aspects of Merkle Trees, a fundamental component in blockchain architectures, such as Ethereum. We critically examine the susceptibility of Merkle Trees to hash collisions, a potential vulnerability that poses significant risks to data security within blockchain systems. Despite their widespread application, the collision resistance of Merkle Trees and their robustness against preimage attacks have not been thoroughly investigated, leading to a notable gap in the comprehensive understanding of blockchain security mechanisms. Our research endeavors to bridge this gap through a meticulous blend of theoretical analysis and empirical validation. We scrutinize the probability of root collisions in Merkle Trees, considering various factors such as hash length and path length within the tree. Our findings reveal a direct correlation between the increase in path length and the heightened probability of root collisions, thereby underscoring potential security vulnerabilities. Conversely, we observe that an increase in hash length significantly reduces the likelihood of collisions, highlighting its critical role in fortifying security. The insights garnered from our research offer valuable guidance for blockchain developers and researchers, aiming to bolster the security and operational efficacy of blockchain-based systems., Comment: 4 figures

Blockchain technology has emerged as a revolutionary tool in ensuring data integrity and security in digital transactions. However, the current approaches to data verification in blockchain systems, particularly in Ethereum, face challenges in terms of efficiency and computational overhead. The traditional use of Merkle Trees and cryptographic hash functions, while effective, leads to significant resource consumption, especially for large datasets. This highlights a gap in existing research: the need for more efficient methods of data verification in blockchain networks. Our study addresses this gap by proposing an innovative aggregation scheme for Zero-Knowledge Proofs within the structure of Merkle Trees. We develop a system that significantly reduces the size of the proof and the computational resources needed for its generation and verification. Our approach represents a paradigm shift in blockchain data verification, balancing security with efficiency. We conducted extensive experimental evaluations using real Ethereum block data to validate the effectiveness of our proposed scheme. The results demonstrate a drastic reduction in proof size and computational requirements compared to traditional methods, making the verification process more efficient and economically viable. Our contribution fills a critical research void, offering a scalable and secure solution for blockchain data verification. The implications of our work are far-reaching, enhancing the overall performance and adaptability of blockchain technology in various applications, from financial transactions to supply chain management., Comment: 18 pages, 7 figures

Information dissemination is a fundamental and frequently occurring problem in large, dynamic, distributed systems. In order to solve this, there has been an increased interest in creating efficient overlay networks that can maintain decentralized peer-to-peer networks. Within these overlay networks nodes take the patterns of small world networks, whose connections are based on proximity. These small-world systems can be incredibly useful in the dissemination and lookup of information within an internet network. The data can be efficiently transferred and routing with minimal information loss through forward error correct (FEC) and the User Datagram Protocol (UDP). We propose a networking scheme that incorporates geographic location in chord for the organization of peers within each node's partitioned key space. When we combine this with a proximity-based neighborhood set {based on the small world structure} we can mimic the efficient of solutions designed to solve traditional small-world problems, with the additional benefit of resilience and fault-tolerance. Furthermore, the routing and address book can be updated based on the neighborhood requirements. The flexibility of our proposed schemes enables a variety of swarm models, and agents. This enables our network to as an underlying networking model that can be applied to file-sharing, streaming, and synchronization of networks.

We propose and define a recursive Merkle structure with q-mercurial commitments, in order to create a concise B-Merkle tree. This Merkle B-Tree builds on previous work of q-ary Merkle trees which use concise, constant size, q-mercurial commitments for intranode proofs. Although these q-ary trees reduce the branching factor and height, they still have their heights based on the key length, and are forced to fixed heights. Instead of basing nodes on q-ary prefix trees, the B Merkle Tree incorporates concise intranode commitments within a self-balancing tree. The tree is based on the ordering of elements, which requires extra information to determine element placement, but it enables significantly smaller proof sizes. This allows for much lower tree heights (directed by the order of elements, not the size of the key), and therefore creates smaller and more efficient proofs and operations. Additionally, the B Merkle Tree is defined with subset queries that feature similar communication costs to non-membership proofs. Our scheme has the potential to benefit outsourced database models, like blockchain, which use authenticated data structures and database indices to ensure immutability and integrity of data. We present potential applications in key-value stores, relational databases, and, in part, Merkle forests.