Low-Complexity Fairness-Oriented Secure Beamforming Design for CAPA-Based Systems

This letter addresses fairness-aware secure beamforming in continuous-aperture array (CAPA)-based multi-user systems. While sum-secrecy-rate (SSR) maximization favors strong users and max-min secrecy-rate (MMSR) optimization is computationally demanding, we adopt the geometric mean secrecy rate (GMSR) as a smooth fairness-oriented alternative. Unlike existing Fourier-discretized methods, the proposed calculus-of-variations (CoV) approach directly optimizes continuous-aperture currents, yielding an integration-free, closed-form, and low-complexity solution with guaranteed convergence. Simulation results show that the proposed CoV-based GMSR algorithm achieves up to 145% higher SSR than MMSR method and over 1300% higher minimum secrecy rate (MSR) than the state-of-the-art SSR-Sun, while operating with orders-of-magnitude lower computational complexity than MMSR.