Secure Protected Regions (CPR) in Near-Field Systems

• CPR is a method that defines a protected zone around the receiver using precise geometric modeling and beamfocusing to counter eavesdropping.
• It employs advanced waveform design and artificial noise to optimize worst-case secrecy rates in near-field multiple-antenna systems.
• Practical implementations via SGDA and Equal-SINRs algorithms, along with full-duplex Bob, enable both physical and virtual protected zones for robust security.

A Secure Protected Region (CPR), with particular reference to receiver-centered protected zones in near-field beamfocusing, is a physical or virtual domain established around a legitimate receiver (“Bob”) wherein eavesdroppers (“Eve”) are either physically excluded or rendered effectively powerless to compromise the secrecy rate of wireless transmissions. In near-field multiple-antenna systems, such as those using ultra-large planar arrays (UPAs), the CPR paradigm exploits spatial geometry and advanced waveform design to enforce strong secrecy guarantees, even against worst-case adversarial positioning. The CPR concept and its algorithmic instantiation underpin a new class of physical-layer security strategies optimized for scenarios where conventional large-scale path loss and far-field beam steering are insufficient to repel closely situated eavesdroppers (Liu et al., 26 May 2025).

1. System Architecture and CPR Definition

The CPR is formally characterized as a spherical region of radius $r_p$ centered at the receiver Bob's known location $p_B$. The secure communications model assumes a large $N$-element UPA at Alice (the transmitter) and single-antenna Bob and Eve. The key geometric construct is the protected zone,

$Z = \{p \mid \|p - p_B\| < r_p\}$

where Eve is precluded from entering $Z$ either by physical constraints or, in virtual implementations, by active jamming and interference management. Line-of-sight (LoS) near-field channels are considered, requiring precise 3D geometric modeling for both $h_B$ (Alice-Bob) and $h_E$ (Alice-Eve), capturing the amplitude decay and phase rotation for each array element. The transmit signal is $x = ws + z$ where $w$ is the analog beamfocusing vector, $s$ the unit-variance information symbol, and $z$ artificial noise (AN), covariance $V \succeq 0$, such that $h_B^H V h_B = 0$ (nulling AN at Bob). The total transmit power is constrained: $\mathrm{Tr}(w w^H + V) \leq P_A$ (Liu et al., 26 May 2025).

2. Secrecy Rate Metrics and Max-Min Formulation

The secrecy rate framework considers both Bob’s achievable rate, $C_B(w) = \log_2(1 + |h_B^H w|^2/(σ_B^2))$, and Eve’s rate at arbitrary position $p_E$, $$C_E(w,V; p_E) = \log_2(1 + |h_E^H w|^2/(h_E^H V h_E + σ_E^2))$$. Instantaneous secrecy rate at $p_E$ is $R_s(w,V; p_E) = [C_B(w) - C_E(w,V; p_E)]^+$.

The critical performance metric is the worst-case secrecy rate,

$$R_s^\mathrm{WC} = \min_{p_E: \|p_E - p_B\| \geq r_p} R_s(w, V; p_E),$$

which underpins the max–min optimization problem:

$$\max_{w, V \succeq 0} \ \min_{p_E: \|p_E - p_B\| \geq r_p} \left\{ C_B(w) - C_E(w, V; p_E) \right\} \quad \text{s.t.} \quad \mathrm{Tr}(w w^H + V) \leq P_A, \ h_B^H V h_B = 0.$$

This design ensures robust secrecy against any external eavesdropper with unknown location outside the protected region (Liu et al., 26 May 2025).

3. Algorithmic Solutions: Synchronous Descent-Ascent and Equal-SINRs

Two principal algorithmic approaches are established:

• Synchronous Gradient Descent-Ascent (SGDA): Handles the inherently non-convex, non-concave (NCNC) and NP-hard max–min secrecy rate problem. The procedure involves parametrizing $w$ by focal point $p_F$ along the Alice–Bob axis, fixing $P_A \to P_\mathrm{TX}$, and using a power-split parameter $\phi$. Projected gradient ascent maximizes over $(\phi, p_F)$, while the inner minimization over $p_E$ (Eve position) invokes an augmented Lagrangian method with synchronized line search (Armijo rule). Candidates for Eve’s position are tracked near the protected-zone boundary to approximate the global minimum.
• Equal-SINRs Solution: This low-complexity approach leverages the near-symmetry of the worst-case Eve positions—$p_1 = (1-r_p/\|p_B\|)p_B$, $p_2 = (1 + r_p/\|p_B\|)p_B$—and reduces the problem to maximizing the minimum secrecy rate over $\{p_1, p_2\}$. For each $p_F$, closed-form splitting coefficients $\phi_1, \phi_2$ are computed, upper-bounding the rate at $p_F$, and the optimal focal point $p_F^*$ is selected. The final step chooses $\phi^*$ to equalize the SINRs at both candidate Eve positions, forming $w$ and $V$ accordingly. The computational complexity is $O(N^2 + C_F N)$, offering performance within 1–2% of SGDA and 10x to 100x faster execution in typical parameter regimes (Liu et al., 26 May 2025).

Approach	Optimization Domain	Key Characteristics
SGDA	$(\phi, p_F)$ (max), $p_E$ (min)	NP-hard, high accuracy
Equal-SINRs	$p_F^*$, $\phi^*$ over two Eve positions	Low complexity, near-optimal

The Equal-SINRs method is particularly suitable for real-time implementation in practical near-field secure systems.

4. Extensions: Virtual Protected Zones with Full-Duplex Bob

When physical enforcement of a protected region is infeasible, a virtual protected region is established via full-duplex Bob. Bob emits artificial noise $u \sim \mathcal{CN}(0, P_B)$ while receiving, encountering residual self-interference $u_R \sim \mathcal{CN}(0, \rho P_B)$ (where $\rho$ models suppression). The effective rates are $$\hat{C}_B = \log_2(1 + |h_B^H w|^2/(\rho P_B + σ_B^2))$$ and

$$\hat{C}_E = \log_2\left(1 + \frac{|h_E^H w|^2}{h_E^H V h_E + \alpha_{BE}^2 P_B + σ_E^2}\right),$$

with $\alpha_{BE} = (2κ d_{BE})^{-1}$, $d_{BE} = \|p_E - p_B\|$. The corresponding max–min optimization seeks $[ \hat{C}_B - \hat{C}_E ]^+$ over all $p_E \in \mathbb{R}^3$ under the same transmit constraints, using heuristic 1D ray search to locate “most-harmful” Eve points for tractability.

The effect of Bob’s AN is to “push” the worst-case Eve position away from Bob, thereby inducing a virtual protected radius $r_v$ where $\|p_{E,\min} - p_B\|=r_v$. Increasing $P_B$ grows $r_v$ until marginal gains abate, particularly as beam-nulling becomes negligible at greater distances (Liu et al., 26 May 2025).

5. Numerical Insights and Implementation Guidelines

Empirical results confirm several design principles. For example, in a 28 GHz, 128×128 UPA, and $P_{TX}=5$ dBm, $\sigma^2=-75$ dBm:

• At $r_p \approx 1$ m, both SGDA-Maximin and Equal-SINRs achieve $R_s \approx 0.8$–$1$ bps/Hz.
• Increasing $r_p$ to $4$ m rapidly boosts $R_s$ to ≈2 bps/Hz.
• Allocating $20$–$30\%$ of Alice’s power to AN ($\phi < 1$) yields substantial gain at small $r_p$, but the benefit diminishes for $r_p \geq 3$ m.
• The Equal-SINRs design tracks within $1$–$2\%$ of SGDA but offers much faster computation.

For virtual protected zones, with $\rho=10^{-8}$ (80–90 dB self-interference cancellation) and $P_B \approx 10$ dBm, $R_s$ versus $P_B$ is unimodal with optimal AN power. Beyond $25$ dBm, Bob’s SI dominates, reducing $R_s$. The induced $r_v$ scales with $P_B$ but saturates once Eve is sufficiently remote (Liu et al., 26 May 2025).

The following table summarizes key design and implementation steps:

Step	Procedure	Applicable Scenario
1	Choose $r_p$	Physical CPR
2	Precompute $p_1$, $p_2$	--
3	Optimize $p_F^*$, $\phi^*$	Equal-SINRs/SGDA
4	Form $w$, $V$	All
5	Select $P_B$ for FD-Bob	Virtual zone
6	Transmit $x = w s + z$, $u \sim \mathcal{CN}(0, P_B)$	Physical/Virtual

6. Practical Considerations and Recommendations

Several actionable guidelines arise:

• Even a modest physical protected-zone radius ($1$–$2$ m) materially improves near-field secrecy performance.
• At $r_p \leq 2$ m, allocate $10$–$30\%$ of Alice’s transmit power to AN; decrease as $r_p$ increases.
• If a physical protected region is infeasible, moderate AN at Bob ($P_B \approx 10$ dBm) with $\sim$80 dB self-interference cancellation emulates a virtual protected region ($r_v \approx 2$–$3$ m).
• Low-complexity Equal-SINRs optimization is recommended for real-time secure communications implementations.

A plausible implication is that the CPR concept, whether enforced physically or virtually, provides a systematic means to achieve robust, worst-case secrecy rates in advanced near-field wireless systems, extending the security envelope beyond what is possible with far-field or purely coding-based approaches (Liu et al., 26 May 2025).