Average Secrecy Rate Maximization

• Average Secrecy Rate Maximization is a framework that optimizes secure wireless transmission by maximizing the expected capacity difference between bona fide and eavesdropping channels.
• It leverages advanced optimization techniques such as block coordinate descent, successive convex approximation, and convex relaxations to address nonconvexity in resource allocation and beamforming.
• Applications span MIMO, IRS-assisted, vehicular, and jamming networks, offering practical design guidelines and computational strategies for enhancing physical-layer security.

The average secrecy rate maximization problem addresses the secure transmission of information in wireless networks under physical-layer security constraints, aiming to maximize the expected secrecy rate in presence of eavesdroppers, fading channels, and practical resource limitations. This problem arises in diverse architectures, including rotatable antenna systems, MIMO wiretap channels, millimeter-wave IRS-assisted networks, untrustworthy relaying networks, vehicular communications, and cooperative networks with mobile jammers. The core objective is to optimize transmission, beamforming, jamming, or system geometry such that the average (ergodic) secrecy rate—defined as the expected difference between legitimate and eavesdropper channel capacities—is maximized over fading and/or spatial realizations.

1. Problem Formulation and Channel Models

In the canonical setting, the average secrecy rate is defined as

$R_s^\mathrm{avg} = \mathbb{E}[C_b - C_e]^+,$

where $C_b$ and $C_e$ denote the legitimate and eavesdropper channel rates, respectively, and expectation is taken over fading realizations or system states. In rotatable antenna-assisted secure communications, the system comprises a single-antenna base station (BS) with a rotatable antenna transmitting to a legitimate user (Bob) at position $q_b \in \mathbb{R}^3$, while an eavesdropper (Eve) at $q_e \in \mathbb{R}^3$ tries to intercept. The rotatable antenna's boresight is parameterized by a deflection vector $\theta = [\theta_z, \theta_a]^T$ or equivalently by an adjustment factor $\alpha$, ensuring the boresight vector lies on the line through $q_e$ and $q_b$ (Jiang et al., 22 Nov 2025).

The channel model typically involves Rician fading: $$h_i(\alpha) = \sqrt{L_i G_i(\alpha)} u_i, \quad L_i = \zeta_0 \|q_i\|^{-\beta_i}, \quad G_i(\alpha) = G_0 \cos \epsilon_i,$$ where $u_i$ models the Rician component, $\epsilon_i$ depends on geometric alignment, and SNR is $\gamma_i(\alpha) = P |h_i(\alpha)|^2 / \sigma^2$.

MIMO wiretap channels generalize this to multiple antennas, where the secrecy rate is

$$R_s(Q) = \log_2 \det(I + H_b Q H_b^H) - \log_2 \det(I + H_e Q H_e^H).$$

For practical multi-carrier MIMO systems, secrecy rate is computed per subcarrier, and auxiliary helpers or jamming nodes may augment the model (Yang et al., 2018, Kalogerias et al., 2013).

IRS and vehicular networks introduce additional beamforming design variables and optimization over resource assignment, subject to power and bandwidth constraints (Rafieifar et al., 2020, Farooq et al., 31 Jan 2024).

2. Optimization Problem Structure and Nonconvexity

The secrecy rate maximization is typically nonconvex due to the difference-of-concave (DC) structure in the objective ($\log\det$ terms for MIMO, or difference of $\log_2(1+\mathrm{SINR})$ terms). In rotatable antenna systems, one maximizes

$$R_s(\alpha) = \mathbb{E}[C_b(\alpha) - C_e(\alpha)]^+ = \int_0^\infty \int_0^x \log_2\left(\frac{1 + \gamma x}{1 + \gamma y}\right) f_b(x;\alpha) f_e(y;\alpha) dy dx,$$

over $\alpha \in [1, \alpha_\mathrm{max}]$ (where $\alpha_\mathrm{max}$ is set by antenna geometry) (Jiang et al., 22 Nov 2025).

In multi-carrier MIMO settings with full-duplex jamming, the maximization is jointly over transmit and jamming covariances under sum-power constraints. Convexification is often achieved by block coordinate descent (BCD), which partitions variables and convexifies each block by auxiliary variable introduction, linearization, or semidefinite relaxation (Yang et al., 2018).

For resource allocation in vehicular networks, the maximization of sum secrecy rate over transmit powers and RB assignments leads to large-scale mixed-integer nonconvex programs, typically solved by successive convex approximation (SCA) or first-order iterative methods (FISTA) (Farooq et al., 31 Jan 2024).

3. Analytical Properties and Solution Methods

The quasi-concavity of the secrecy rate with respect to controllable parameters (e.g., antenna adjustment factor $\alpha$) is instrumental in enabling efficient global optimization. In rotatable antenna systems, it is proven that $R_s(\alpha)$ is quasi-concave on the feasible interval, allowing optimal selection by bisection search (Jiang et al., 22 Nov 2025). The proof utilizes joint PDF reparameterization and monotonicity arguments for channel alignment metrics.

For MIMO wiretap channels, DC programming and eigenvalue-based alternating optimization (e.g., POTDC) yield stationary points of the secrecy rate function (Khabbazibasmenj et al., 2014). In multi-carrier systems, BCD alternates covariance updates with auxiliary matrix inversion, leveraging convex subproblems for convergence to KKT points (Yang et al., 2018).

In vehicular networks, SCA transforms bilinear secrecy rate constraints into convex subproblems by introducing slack variables and quadratic or second-order cone (SOC) relaxations. FISTA offers a lower-complexity alternative by applying projected gradient methods to a smoothed surrogate objective (Farooq et al., 31 Jan 2024).

The closed-form optimality conditions are available in some specific setups, such as power allocation in secure directional modulation networks (quadratic equations in the power allocation factor $\beta$) and under NSP beamforming where the relevant stationary condition reduces to polynomial roots in $\beta$ (Wan et al., 2018).

4. Special Cases, Extensions, and Numerical Investigations

Closed-form near-optimal solutions exist in deterministic channel limits. For pure LoS, the optimal boresight angle in rotatable antenna systems is explicitly characterized, and the impact of antenna adjustment can be highly pronounced, particularly when geometric alignment nullifies eavesdropper gain (Jiang et al., 22 Nov 2025).

Power allocation in MIMO wiretap channels with statistical CSI is efficiently handled via GSVD, reducing the problem to concave maximization over parallel scalar-wiretap channels, and for finite-alphabet input, an additional upper-bound per-variable achieves full secrecy rate at high transmit powers (Vishwakarma et al., 2014).

In mobile jammer-assisted networks, the control of helper motion via decentralized gradient laws maximizes the average secrecy rate with minimal helper count and transmit power (Kalogerias et al., 2013). In hardware-impaired relaying scenarios, SPCA-based iterative convex approximation and initialization routines ensure high feasibility and convergence rates, while deep learning-based surrogates facilitate near-optimal solution prediction in large-scale networks with substantial runtime reductions (Bastami et al., 2021).

IRS-assisted mmWave networks utilize alternating SDR methods for joint optimization of active and passive beamforming, and simulation results demonstrate monotonic secrecy rate improvements with IRS element number and BS antennas, with diminishing gains for additional IRSs (Rafieifar et al., 2020).

5. Computational Complexity and Convergence Analysis

Solution complexity is dominated by the underlying convex program solvers, such as MAX-DET subsolver in block coordinate descent ($O(n^3)$ per iteration for MIMO), SDP solvers in IRS-assisted systems ($O(M^{3.5} + L(N+1)^{3.5})$ per iteration), and the number of gradient evaluations or expansion points in first-order methods like FISTA is $O(KM)$ per iteration (Yang et al., 2018, Rafieifar et al., 2020, Farooq et al., 31 Jan 2024).

Iterative schemes such as SCA or BCD typically converge in 10–20 iterations for moderate-size problems, with convergence guarantees hinging on monotonicity and boundedness of the secrecy rate objective. Deep learning surrogates produce solution predictions with sub-millisecond latency, offering several orders of magnitude speed-up compared to iterative optimization (Bastami et al., 2021).

6. Practical Implications and Design Guidelines

Physical-layer secrecy rate maximization in average sense enables secure transmission under realistic channel estimation and adjustment times. The use of only positional information in closed-form rotatable antenna designs provides system robustness when channel conditions vary slower than mechanical reorientation (Jiang et al., 22 Nov 2025). Full-duplex jamming and helper mobility exploit spatial and frequency diversity, substantially enhancing secrecy rates, especially under strong eavesdropper channels (Yang et al., 2018, Kalogerias et al., 2013).

Optimizing transmit and jamming power, resource allocation, and beamforming variables increases secrecy rates and resource efficiency, providing substantial gains over naive or static design strategies, particularly in high-SNR and multi-antenna regimes (Rafieifar et al., 2020, Wan et al., 2018).

Efficient algorithmic implementation, including convex relaxations, alternating optimization, and machine-learning prediction, is critical for real-time network operation in high-mobility environments such as vehicular systems. SCA offers superior optimality at moderate computational cost, while FISTA enables rapid adaptation at slight optimality loss (Farooq et al., 31 Jan 2024).

7. Open Challenges and Future Directions

Average secrecy rate maximization remains challenging in high-dimensional and highly dynamic network environments, especially when statistical or partial CSI is available, or finite-alphabet coding is considered. Robustness to CSI uncertainty, scalability to massive MIMO or large IRS deployments, integration with resource management and topology control, and joint optimization across multi-hop or cooperative relay networks are current subjects of research.

Sample-average approximation and analytical bounding approaches continue to be necessary for ergodic-CSI settings, while low-complexity surrogates such as deep learning are increasingly viable for large-scale deployments requiring near-instantaneous resource re-allocation (Khabbazibasmenj et al., 2014, Bastami et al., 2021).

In sum, average secrecy rate maximization forms the backbone of physical-layer security designs across modern wireless architectures, providing both theoretical guarantees and practical frameworks for secure communication under adversarial and uncertain environments.