Brownian motion models: cryptographic applications, capabilities, and limitations

Brownian motion (BM) is a stochastic model that has been extensively studied in physics, finance, and engineering. However, its potential use in cryptographic applications remains underexplored. This paper presents a comprehensive review of the capabilities, limitations, and cryptographic properties of various BM models, including the Wiener process, geometric BM, fractional BM, Ornstein–Uhlenbeck process, multidimensional BM, and reflected BM. We reviewed the mathematics of these models, simulate their random evolutions, and compare their cryptanalytic properties. A comparison of these sources highlights unique characteristics that can provide cryptographic resilience, including long-range dependence, multidimensional modeling of noise, and constraints on randomness. We also describe the main limitations and potential weaknesses of each model. This paper addresses gaps in the application of stochastic process to cryptographic design and provides a foundational guideline for the continued development of secure systems based on Brownian dynamics.