Multi-User Key Rate Optimization for Near-Field Extremely Large-Scale Antenna Array Communications

Extremely large-scale antenna arrays (ELAA) require near-field spherical wave modeling due to the substantial increase in the number of antennas, which introduces new spatial dimensions to physical layer key generation (PLKG). We investigate multi-user PLKG in near-field environments, where a base station with an ELAA simultaneously generates secret keys with multiple users. We derive an analytical expression for the key rate (KR). By utilizing spatial dimensions of distance and angle in near-field environments, we apply eigenvalue decomposition and singular value decomposition to design precoding matrices to reduce interference among user equipments (UEs) and extract uncorrelated subchannels. Given that the KR is non-convex, we approximate it and optimize the precoding matrix to increase the KR. After precoding design, the KR depends on the transmit power allocated to the subchannels. Two optimization problems are formulated to further optimize transmit power allocation. The first problem focuses on maximizing the sum KR. We apply the Lagrange multiplier method to determine the optimal power allocation variables by searching the Lagrange multiplier. To reduce computational complexity, a supervised feedforward neural network (FNN) is designed to capture the relationship between the power allocation variables and the Lagrange multiplier. The second optimization problem focuses on KR fairness. By introducing a slack variable that is smaller than the KRs of all users, we use the CVX toolbox to find optimal power allocation variables that maximize this slack variable. To further reduce complexity, the Lagrange multiplier method offers an analytical solution for power allocation variables in terms of Lagrange multipliers determined by the slack variable in the high-power case. We employ a bisection algorithm to find the slack variable. Furthermore, we propose an FNN to map transmit power to the slack variable. Simulations demonstrate that the proposed methods efficiently leverage near-field effects for multi-user PLKG, reducing pilot overhead.