Max-Min Secrecy Rate Optimization for Secure ISAC Networks: Global Optimization and Low-Complexity Algorithm

Source: arXiv:2606.13582 · Published 2026-06-11 · By Thanh-Nha To, Trung Quang Pham, Dang Y Hoang, Hoang-Lai Pham, Tuan Anh Pham

TL;DR

This paper addresses the problem of secure multi-user integrated sensing and communication (ISAC) networks where untrusted sensing users (SUs) may eavesdrop on confidential information intended for legitimate communication users (CUs). It formulates a max-min secrecy rate optimization problem to ensure fairness among communication users under transmit power and sensing quality constraints, the latter characterized by beampattern matching error. The problem is highly non-convex due to the secrecy rate definitions and sensing constraints. The authors propose a global optimization framework combining semidefinite relaxation (SDR) and a branch-and-bound (BB) algorithm with convex relaxations to obtain globally optimal beamforming solutions within a prescribed accuracy. To provide a practical low-complexity alternative, they also develop a successive convex approximation (SCA)-based iterative algorithm that converges to locally optimal solutions while significantly reducing computational complexity. Numerical experiments under multi-user Rician fading channels demonstrate that the BB algorithm achieves the global optimum, which serves as a benchmark, while the SCA method obtains near-optimal secrecy rates with much lower complexity, thus offering a practical solution for large-scale secure ISAC systems.

Key findings

• The proposed branch-and-bound (BB) algorithm converges to a global ϵ-optimal solution with provable upper and lower bounds on secrecy rate within a prescribed tolerance (ϵ).
• The low-complexity successive convex approximation (SCA) algorithm converges significantly faster than BB, reaching near-global-optimal secrecy performance in around 10 iterations versus over 100 iterations for BB in K=3 CU scenario (Fig 2, 3).
• As the number of communication users (CUs) grows, BB complexity and convergence time grow rapidly (exponentially), while SCA remains efficient and close to optimal, converging within ~10 iterations up to K=4 users (Fig 3).
• Tightening sensing quality (lower max beampattern error) reduces the minimum secrecy rate drastically due to resource trade-offs with sensing, observed across multiple CU numbers (Fig 4).
• Increasing the number of BS antennas from 10 to 32 increases minimum secrecy rate and mitigates sensing-secrecy trade-offs, allowing high secrecy rates even under strict sensing error constraints (Fig 5).
• The semidefinite relaxation (SDR) based formulation guarantees rank-one solutions for beamforming matrices without loss of optimality (Theorem 1).
• Convex envelope relaxation of non-convex exponential constraints enables effective bounding in BB method for global optimization.
• The SCA algorithm ensures monotonic improvement with convergence guarantees, offering a polynomial time complexity per iteration versus exponential in BB.

Threat model

The adversaries are untrusted sensing users (SUs) within the ISAC network that can passively eavesdrop on confidential communication signals intended for legitimate communication users (CUs). They are assumed to have perfect knowledge of their own channels to the BS and employ optimal detection strategies. However, they cannot actively alter the communication environment or signals, nor can they collude or cooperate beyond independent eavesdropping. The system aims to design transmit beamforming to maximize secrecy under worst-case interception from any of these SUs.

Methodology — deep read

1. Threat Model & Assumptions: The adversary consists of multiple untrusted sensing users (SUs) who act both as sensing targets and potential eavesdroppers capable of intercepting the BS's transmissions to legitimate communication users (CUs). The BS uses physical layer security to mitigate these threats, designing transmit beamforming to maximize the minimum secrecy rate while meeting sensing quality and power constraints. The adversaries can eavesdrop but cannot alter signals or perform active attacks.

2. Data: The authors simulate an ISAC system with a base station (BS) having NB=16 antennas, serving K=3 single-antenna CUs and sensing J=3 targets modeled with line-of-sight and Rician fading channels (Rician factor 3 dB). Users and sensing targets are placed geographically as specified. Noise is set to -90 dBm. 100 channel realizations are simulated for performance evaluation.

3. Architecture/Algorithm: The joint beamforming design problem maximizes the minimum per-user secrecy rate. The secrecy rate for each CU is defined as the difference between user achievable rate and the maximum eavesdropping rate among SUs, clipped to zero if negative. Sensing quality is quantified by the squared error of beampattern matching with desired directional energy distributions. The optimization variables include the covariance matrices of artificial noise and user beamformers and a scaling factor for sensing.

The non-convex problem involves rank-one constraints on beamforming matrices, log functions in rate expressions, and non-convex sensing constraints. It is reformulated using auxiliary variables and semidefinite relaxation (dropping rank-one constraints). Further transformations introduce auxiliary variables a,b to handle concave logarithms. Non-convex constraints are convexified via the convex envelope relaxations of exponential functions, yielding a convex problem suitable for bounding.

4. Training/Computation: The global solution uses a branch-and-bound (BB) algorithm that recursively partitions the box of auxiliary variables a,b, solving convex relaxations to get upper bounds and constructing feasible solutions for lower bounds. The BB terminates when the upper-lower gap falls below tolerance ϵ. Complexity grows exponentially with problem dimension, especially number of users.

Alternatively, a low-complexity successive convex approximation (SCA) algorithm linearizes exponential constraints via first-order Taylor approximations around previous iterates. It solves a sequence of convex SDPs updating linearization points iteratively until convergence criteria (objective improvement less than ϵ) is met. This provides a local optimum with polynomial time complexity per iteration.

5. Evaluation Protocol: Metrics include minimum secrecy rate (bps/Hz) among all CUs. They compare BB upper and lower bounds with SCA solutions over 100 channel realizations. Scenarios vary number of CUs, sensing error constraints, and number of BS antennas. Relative error to global bounds measures approximation quality. Convergence rates and iteration counts are analyzed.

6. Reproducibility: Algorithms are implemented using CVXPY convex solvers. No explicit code or dataset release is mentioned, but simulation parameters and channel models are described in detail for reproducibility. Closed form problem reformulations and algorithm pseudocode are provided.

Concrete Example: For a system with NB=16 antennas, K=3 CUs, and J=3 SUs under Rician fading, the BB algorithm initializes auxiliary variable bounds based on maximum transmit power and channel gains. It then partitions this space into sub-boxes, solving convex relaxation (17) within each. Feasible solutions are constructed by adjusting auxiliary variables using logarithm and exponential transformations to maintain bounds. Iterations refine these boxes until the secrecy rate gap of upper and lower bounds is below 10^-2 bps/Hz. The SCA algorithm solves convex approximations of problem (38) starting from initial points derived from the same bounds, updating linearized constraints until the secrecy rate converges within the same tolerance but in fewer than 20 iterations.

Technical innovations

• A novel global optimization framework combining semidefinite relaxation, convex envelope relaxations, and branch-and-bound to solve max-min secrecy rate beamforming in secure multi-user ISAC.
• A low-complexity successive convex approximation algorithm that provides high-quality local solutions with convergence guarantees and polynomial complexity, suitable for large-scale ISAC systems.
• Proof of relaxation tightness guaranteeing rank-one optimal solutions for the SDR formulation despite problem non-convexity.
• Derivation and use of convex envelopes of exponential sets for constructing tight convex relaxations enabling efficient bounding in BB.

Baselines vs proposed

• Branch-and-bound (BB) algorithm: achieves global optimum secrecy rate benchmark (upper and lower bounds converge to same value).
• Successive Convex Approximation (SCA): minimum secrecy rate within 1% of BB global optimum while requiring approximately 10× fewer iterations (10 versus ~110 iterations for K=3).
• BB convergence: relative error under 10^-2 in fewer than 100 iterations for K=3; exceeds 103 iterations for K=4.
• SCA convergence: relative error under 10^-2 within 10 iterations even for K=4.
• Sensing beam pattern constraint relaxation from -26dB to 0dB increases min secrecy rate from near 0 to saturation values, demonstrating trade-off.
• Increasing number of BS antennas from 10 to 32 raises min secrecy rate by up to 14 bps/Hz and reduces sensing-secrecy trade-off.

Limitations

• Computational complexity of global BB approach becomes prohibitive as number of communication users (CUs) increases beyond 4 due to exponential branching.
• The low-complexity SCA algorithm provides only local optima, lacking global optimality guarantees.
• Simulations use relatively small-scale setups (up to 8 users, 32 antennas); performance and scalability for massive ISAC arrays remain untested.
• Assumes perfect knowledge of channel state information (CSI) and target locations; robustness under imperfect or delayed CSI is not studied.
• Eavesdroppers are passive and only listen; active adversaries or jammers are not considered.
• No real-world experiments or code release limit immediate reproducibility or practical validation.

Open questions / follow-ons

• How does imperfect or partial channel state information (CSI) affect the secrecy performance and optimization outcomes?
• Can the proposed framework be extended to active adversarial models involving jamming or signal injection?
• How does mobility and time-variation of user and target locations impact beamforming design and secrecy guarantees?
• What are the trade-offs and performance under more heterogeneous ISAC environments with more users, antennas, and multi-cell deployments?

Why it matters for bot defense

This work is directly relevant to bot-defense and CAPTCHAs insofar as it tackles physical-layer security and fairness in wireless systems that integrate sensing and communication, scenarios that could underpin sensing-based authentication or anomaly detection mechanisms. The max-min secrecy rate formulation addresses fairness among users in adversarial environments, analogous to ensuring robust verification under various threat models. The global and low-complexity algorithms provide design tools for securing systems where eavesdropping is a concern, which can inspire new defenses against automated attacks that exploit sensing signals or side channels.

Bot-defense engineers could borrow the branch-and-bound convex relaxation framework to design provably secure physical layer schemes, or leverage the low-complexity SCA approach for real-time system deployment that balances security and sensing trade-offs. This research emphasizes the importance of joint optimization across multiple objectives (security, power, sensing quality), highlighting the complexities of defending integrated systems where classical approaches may fail. While not a CAPTCHA design itself, the underlying principles of fairness, secrecy maximization under adversarial constraints, and algorithmic efficiency provide valuable insights for developers of multi-factor, signal-based bot defenses and detection algorithms.

Cite

@article{arxiv2606_13582,
  title={ Max-Min Secrecy Rate Optimization for Secure ISAC Networks: Global Optimization and Low-Complexity Algorithm },
  author={ Thanh-Nha To and Trung Quang Pham and Dang Y Hoang and Hoang-Lai Pham and Tuan Anh Pham },
  journal={arXiv preprint arXiv:2606.13582},
  year={ 2026 },
  url={https://arxiv.org/abs/2606.13582}
}

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• arXiv:2606.13582 (abstract)
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