This study presents the development of a novel three-dimensional chaotic system incorporating a hyperbolic sine function, characterized by its chaotic dynamics and applications in pseudo-random number generation (PRNG) for medical image encryption. Our system exhibiting rotational symmetry under a \(180^{\circ }\) transformation about the z-axis. Dynamical analyses, including bifurcation diagrams and Lyapunov spectrum evaluations, confirm multistability and coexistence of periodic and chaotic attractors. As an engineering application, a PRNG based on the system is implemented for medical image encryption. The PRNG passes all NIST-800-22 statistical tests, achieving high randomness with test results ranging between 0.001 and 1. Image encryption is tested on head and chest images, yielding statistical metrics such as entropy (79.566 and 79.576), contrast (105.163 and 104.860), and encryption quality (28004 and 25800), demonstrating the robustness and reliability of the proposed approach. This work offers a promising tool for secure medical image processing with high computational efficiency.
