The Lethargic Kangaroo: Diffusive Mixing and Coupling Latency in Clustered Step Sets

Hardware implementations of parallel collision search algorithms (Pollard’s Rho and Lambda) often restrict the iteration function’s “step set” to reduce logic area. This paper analyzes the probabilistic cost of “clustered” step sets (e.g., S = {s, s + 1}) where the step variance is minimized. We demonstrate that while such sets maintain high frequency (fmax), they possess a vanishingly small diffusive expansion rate. This leads to stiff trajectories that fail to couple efficiently. We identify the primary failure mode as coupling latency derived from the slow evolution of the relative distance between walkers. Simulation confirms that low-variance step sets increase the expected time-to-collision by 12–15%, a direct loss in wall-clock cryptographic efficiency.