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.

翻译：针对去中心化系统中确保数据完整性的关键挑战，本文深入探讨了Merkle树中数据伪造概率这一研究不足的领域，而Merkle树在区块链与物联网技术中具有核心地位。尽管其应用广泛，目前对这些结构中数据安全概率层面的全面理解仍存在研究空白。本研究旨在通过构建理论框架计算数据伪造概率来填补这一空白，该框架考虑了基于Merkle路径长度与哈希长度的多种场景。研究从推导伪造概率的精确公式出发，进一步针对哈希长度极大情况提出近似方法。实证实验通过模拟不同哈希长度与Merkle路径长度验证了理论模型。结果表明，伪造概率随哈希长度增加而降低，且与更长的Merkle路径呈反比关系。数值分析量化了精确概率与近似概率之间的差异，明确了近似方法的有效应用条件。本研究为优化Merkle树结构以增强区块链与物联网系统的安全性提供了重要见解，实现了计算效率与数据完整性之间的平衡。