Cryptographic analysis of chaotic systems

In computer science, hash functions are elementary operators that convert arbitrary-length inputs into finite-length outputs. We describe a direct analogy between these functions and the trajectories of particles advected by fluid flows, and we show that, when the governing flow is chaotic, hydrodynamic hash functions exhibit statistical properties typically associated with hash functions used for digital cryptography. These include non-invertibility, sensitivity to initial input, and avoidance of collisions—a phenomenon in which two similar inputs produce identical outputs. We show that this analogy originates from the tendency of certain chaotic flows to braid together particle trajectories across space in time in an irreducible manner, and we describe how techniques used to probe the properties of digital hash functions can be used to characterize the properties of flows when only limited observational data is available. Our findings have potential applications in microfluidic proof-of-work systems, as well as for characterizing large-scale transport in ocean flows and living systems.