Cryptographic hashing using chaotic hydrodynamics
Proceedings of the National Academy of Sciences of the United States of America
Fluids may store and manipulate information, enabling complex applications ranging from digital logic gates to algorithmic selfassembly. While controllable hydrodynamic chaos has previously been observed in viscous fluids and harnessed for efficient mixing, its application to the manipulation of digital information has been sparsely investigated. We show that chaotic stirring of a viscous fluid naturally produces a characteristic signature of the stirring process in the arrangement of particles
... in the fluid, and that this signature directly satisfies the requirements for a cryptographic hash function. This includes strong divergence between similar stirring protocols' hashes and avoidance of collisions (identical hashes from distinct stirs), which are facilitated by noninvertibility and a broad chaotic attractor that samples many points in the fluid domain. The hashing ability of the chaotic fluidic map implicates several unexpected mechanisms, including incomplete mixing at short time scales that produces a hyperuniform hash distribution. We investigate the dynamics of hashing using interparticle winding statistics, and find that hashing starts with large-scale winding of kinetically disjoint regions of the chaotic attractor, which gradually gives way to smaller scale braiding of single-particle trajectories. In addition to providing a physically motivated approach to implementing and analyzing deterministic chaotic maps for cryptographic applications, we anticipate that our approach has applications in microfluidic proof-of-work systems and characterizing large-scale turbulent flows from sparse tracer data. nonlinear dynamics | fluid dynamics | encryption | mixing | braiding R ecent experimental work has highlighted the ability of fluids to encode and store information (1, 2), motivating reciprocal inquiry into the role of computational rules in shaping the behavior of fluids in the natural world (3, 4). Such work has established applications in improving complex microfluidic devices and for characterizing large-scale complex flows using sparse data (5, 6), but it has broader implications for understanding constraints that shape active matter and self-assembly schemes (7). At the same time, studying logical operations and algorithmic performance in physical systems allows analytical tools borrowed from physics to be applied to traditional digital information systems, a line of inquiry that traces from Wheeler's original "it from bit" conjecture to Landauer's arguments about the role of physical representation on information processing (8, 9) . A potential new avenue for such inquiries is chaos, which has been widely investigated in digital applications due to the rich statistical structure it affords deterministic (and thus manipulable) dynamical systems (10). While hydrodynamic systems have been shown to exhibit chaos-both ubiquitously in turbulent flows (11, 12) but also unexpectedly in viscous flows via elegant analogies to classical dynamical nonintegrability (13)the implications of chaos for digital fluid physics remain mostly unexplored. Here, we exploit recent advances in the field of chaotic hydrodynamics to show how well-understood properties of chaotic maps can encode information about the underlying flow dynamics into the relative arrangements of advected particles. We show that this operation satisfies all of the properties of cryptographic hash functions that typically appear in digital security applications, including noninvertible compression of arbitrary inputs to fixed-length outputs, strong divergence between the hashes of similar inputs, and resistance to collisions between the hashes of two distinct inputs (14) . We show that these unexpected properties arise naturally from the time scale-dependent dynamics of stirring a viscous liquid, implying potential new analysis techniques and applications at the interface of nonlinear dynamics and cryptography. Model Our cryptographic hashing scheme is based on chaotic advection at low Reynolds number. Given a time-varying flow and a small set of particles being advected, our hash consists of a short digest containing the relative ordering of the particles along one dimension. In order for the hashing scheme to be effective, this digest must be unique to the specific flow, but the original flow itself should not be easily computed from the hash-a property that naturally emerges in chaotic flows. Under our approach, if the flow being studied has a known, finite set of governing parameters (such as jet speeds or stirring rates), then the time-dependent flow itself may be denoted by a discrete sequence of L vectors of parameter values σ, which we refer to as a "stirring protocol" for the flow. The time step between parameter changes is arbitrary and may even be infinitesimal (corresponding to an analog signal); however, we assume that dissipation is large enough (and thus the Reynolds number and inertia are small enough) that the stirring protocol fully and invertibly specifies the dynamics of particles in the flow. We associate the specific stirring protocol σ with a "message" of length L that we wish to encrypt. We then specify M labeled particles at known initial positions and allow the flow to advect these particles for L time steps with the step-wise parameters specified by σ. The final arrangement of these particles is discretized by recording their Significance An essential component of digital communication is hashing, in which a complex piece of information (a document, video, etc.) is mathematically transformed into a unique signature that can later be used to identify the original piece of data. Here, we show that this process bears strong similarity to the chaotic behavior of certain types of flows observed when ordinary fluids mix, such as the stirring of dye into water. We use this analogy between rearranging information and stirring a fluid to construct a fluid-based hash function with comparable properties to traditional algorithms. Our work bears direct relevance to cases in which the physical representation of information affects its transmission, including in microfluidic self-assembly schemes and characterizing complex natural flows.