Linear and Numerical SDoF Bounds of Active RIS-Assisted MIMO Wiretap Interference Channel

The multiple-input multiple-output (MIMO) wiretap interference channel (IC) serves as a canonical model for information-theoretic security, where a multiple-antenna eavesdropper attempts to intercept communications in a two-user MIMO IC system. The secure degrees-of-freedom (SDoF) of an active reconfigurable intelligent surface (RIS)-assisted MIMO wiretap IC is with practical interests but remains unexplored. In this paper, we establish both sum-SDoF lower and upper bounds through linear beamforming conditions and numerical methods. Specifically, our proposed lower bound is derived from transmission scheme design and corresponding solutions to the sum-SDoF maximization problem, formulated by linear integer programming. The solutions to this optimization problem addresses RIS element allocation for leakage and interference cancellation. The proposed upper bound is obtained by solving a nuclear norm minimization problem, leveraging the fact that nuclear norm serves as a convex relaxation of the rank function. For symmetry antenna configurations, we derive a closed-form lower bound. Extensive numerical simulations show that our proposed lower and upper bounds coincide across many antenna configurations, and our proposed lower bound outperforms the existing benchmark.