Analytical modeling of the attack of double costs on a blockchain system with a probabilistic consensus protocol

Abstract

To build safe and reliable distributed decentralized systems using blockchain technology, a continuous chain of blocks is created, the unauthorized modification of which is not allowed by the applied cryptographic mechanisms. This is achieved by using unidirectional, collision resistant and search prototypes of cryptographic functions whose hash values from previous blocks are included in subsequent blocks. As a result, an unauthorized change in at least one bit of data in the previous blocks will be immediately detected. However, in the case of distributed storage of information, there is an additional requirement of synchronization of individual chains of blocks that are stored by various nodes. These and other issues are resolved by applying consensus building mechanisms, through which, after a certain sequence of actions, a continuous sequence of blocks (blockchain chain) becomes the same on all nodes of a decentralized network. This work examines one of the main vulnerabilities of blockchain systems built by consensus with probabilistic completion, namely, a double-spend attack. Based on the model of “independent players”, an analytical expression is obtained for calculating the probability of an attacker's successful double-spending attack. The quantitative values of the probability of a successful attack are given for various abilities of an attacker, a different number of generated blocks and a different race duration. Using computer simulation, the calculated values are experimentally verified. All empirical estimates were obtained with high accuracy (relative error not higher than 1%) and reliability (confidence level of at least 99%). To confirm the adequacy of the results obtained, a comparison of empirical results with theoretical calculations is given.