The discrete logarithm problem is a fundamental challenge in number theory with significant implications for cryptographic protocols. In this paper, we investigate the limitations of gradient-based methods for learning the parity bit of the discrete logarithm in finite cyclic groups of prime order. Our main result, supported by theoretical analysis and empirical verification, reveals the concentration of the gradient of the loss function around a fixed point, independent of the logarithm's base used. This concentration property leads to a restricted ability to learn the parity bit efficiently using gradient-based methods, irrespective of the complexity of the network architecture being trained. Our proof relies on Boas-Bellman inequality in inner product spaces and it involves establishing approximate orthogonality of discrete logarithm's parity bit functions through the spectral norm of certain matrices. Empirical experiments using a neural network-based approach further verify the limitations of gradient-based learning, demonstrating the decreasing success rate in predicting the parity bit as the group order increases.

Event cameras are bio-inspired dynamic vision sensors that respond to changes in image intensity with a high temporal resolution, high dynamic range and low latency. These sensor characteristics are ideally suited to enable visual target tracking in concert with a broadcast visual communication channel for smart visual beacons with applications in distributed robotics. Visual beacons can be constructed by high-frequency modulation of Light Emitting Diodes (LEDs) such as vehicle headlights, Internet of Things (IoT) LEDs, smart building lights, etc., that are already present in many real-world scenarios. The high temporal resolution characteristic of the event cameras allows them to capture visual signals at far higher data rates compared to classical frame-based cameras. In this paper, we propose a novel smart visual beacon architecture with both LED modulation and event camera demodulation algorithms. We quantitatively evaluate the relationship between LED transmission rate, communication distance and the message transmission accuracy for the smart visual beacon communication system that we prototyped. The proposed method achieves up to 4 kbps in an indoor environment and lossless transmission over a distance of 100 meters, at a transmission rate of 500 bps, in full sunlight, demonstrating the potential of the technology in an outdoor environment.