Statistical Indistinguishability in Multi-User Covert Communications Without Secret Information

Abstract

This paper proposes a novel covert communication paradigm in which covertness emerges from network-induced structural uncertainty, eliminating the traditional reliance on pre-shared secret pilots in multi-user cooperative networks. Unlike conventional schemes that create information asymmetry through secret training sequences, we show that structural uncertainty naturally arises from user selection in spatially dispersed networks. Specifically, we consider a public pilot aided system under a worst-case adversarial assumption where Willie possesses full knowledge of all individual channel state information (CSI) but remains uncertain about the active subset of cooperative users. We prove that this selection-induced structural uncertainty renders different transmission states statistically indistinguishable from Willie’s perspective, thereby forcing the optimal detector to reduce to an energy-based test. The proposed framework demonstrates that robust covertness can be achieved without secrecy-based coordination, providing a scalable and practically viable alternative to secret pilot management in future wireless networks.

1. Introduction

Covert communication, also known as low probability of detection (LPD) communication, has emerged as a vital security paradigm designed to conceal the very occurrence of a wireless transmission from a sophisticated warden, Willie [1]. Unlike traditional physical layer security (PLS) techniques that primarily focus on protecting the confidentiality of message content from eavesdroppers [2], covert communication aims to maintain a high detection error probability (DEP) at Willie, thereby hiding the communication event itself. Fundamental studies established the square root law (SRL) [3,4], showing that in an additive white Gaussian noise (AWGN) environment, Alice can transmit at most $\mathcal{O}\left(\sqrt{n}\right)$ bits over n channel uses without being detected. However, the SRL implies that the achievable covert rate per channel use approaches zero as the codeword length increases, which severely limits practical communication efficiency [5].

The square root law was initially established for additive white Gaussian noise (AWGN) channels [3], demonstrating that Alice can transmit at most $\mathcal{O}\left(\sqrt{n}\right)$ covert bits over n channel uses. Subsequent studies extended this foundational result to binary symmetric channels (BSCs) [6], discrete memoryless channels (DMCs) [4,5], and multiple access channels (MACs) [7], confirming the generality of the square root scaling across diverse channel models. While these works established the theoretical underpinnings of covert communication, they also revealed a fundamental limitation: the achievable covert rate per channel use vanishes as the codeword length increases, motivating the search for mechanisms that enable positive covert rates.

Several approaches have been proposed to overcome this limitation by introducing uncertainty at the warden’s detector. Noise uncertainty-based schemes [8] exploit imprecise knowledge of the noise floor to sustain positive covert rates, but their performance degrades as the warden improves its receiver calibration. Channel state information (CSI) uncertainty-based approaches [9,10] leverage channel estimation errors to mask covert transmissions behind a legitimate communication link, where the warden’s imperfect channel knowledge prevents reliable detection. Cooperative jamming methods introduce friendly interference to disrupt the warden’s detection. Single-jammer models [11,12] create uncertainty through random power variation of a dedicated jammer node, while multi-user cooperative jamming [13] employs multiple spatially distributed jammers whose independent activation generates aggregate interference uncertainty. In stochastic geometry frameworks [14], the spatial randomness of interferer locations provides a natural source of uncertainty. Common to all these approaches is the reliance on some form of information that is hidden from the warden, whether it be secret codebooks, secret pilots, or hidden interferer identities.

More recently, multi-user cooperative frameworks have attracted significant attention for their potential to scale covert communication to practical network settings. In [15,16,17], deterministic exact-K user selection is employed to generate cooperative jamming, where secret pilot sequences prevent the warden from estimating the cooperative channels. While effective, the reliance on secret pilots naturally raises a theoretical question: whether covertness can be achieved without any secret information, by exploiting the structural properties of the network itself. Addressing this question not only deepens the theoretical understanding of covert communication but also opens the possibility of complementary designs that do not require secret coordination infrastructure.

In contrast to conventional secrecy-based approaches, this paper investigates a public pilot aided network environment in which Willie possesses the full CSI of all individual links. We show that covertness does not fundamentally require secret information. Instead, structural uncertainty naturally emerges from user selection in spatially dispersed networks. Because cooperative users are located at unequal distances from Willie, each potential link contributes a distinct path-loss component to Willie’s received signal. Although Willie perfectly knows the individual channels, $h_{a,w},h_{1,w},\dots ,h_{M,w}$, through public pilots, he remains uncertain about which subset of users is active. This selection-induced structural uncertainty creates an irreducible ambiguity in Willie’s observation.

The key insight of this work is that spatial dispersion transforms network geometry into a security asset rather than a limitation. The active subset S, where only K out of M users participate, is selected according to legitimate channel conditions toward Bob. As a result, the mapping between observed aggregate energy and specific channel realizations becomes statistically indistinguishable from Willie’s perspective. Even with perfect CSI, Willie cannot reliably infer which users are contributing to the observed signal, and the uncertainty induced by network geometry persists as a fundamental barrier to detection.

This result demonstrates a new design principle for covert communication: covertness can emerge from network-induced structural uncertainty rather than secrecy-based coordination. By leveraging spatial dispersion and user selection, the proposed framework provides a scalable and practically efficient alternative to secret pilot management while maintaining robust covertness under a worst-case adversarial model.

The main contributions of this work are summarized as follows:

• Structural covertness without secret pilots: We depart from the conventional secret pilot paradigm and demonstrate that robust covertness can be achieved in a fully public pilot environment. Even when Willie possesses complete individual channel state information (CSI), network-induced structural uncertainty enables covert communication without secrecy-based coordination.
• Optimal detection under structural uncertainty: We mathematically prove that, despite full CSI knowledge, Willie’s optimal detection strategy reduces to an energy detector due to the uncertainty in the active user subset. Based on this result, we derive a closed-form expression for the optimal transmit power $P_a^{*}$ that satisfies a given covertness constraint $ϵ$.
• Spatial dispersion as a fundamental security resource: We show that spatial dispersion among cooperative users amplifies the structural uncertainty observed at Willie. This geometric diversity increases the variance of aggregate interference, allowing Alice to increase her transmit power while maintaining the required level of covertness.
• Optimal active subset design: Through theoretical analysis and numerical evaluation, we identify an optimal number of active cooperative users $K^{*}$ that maximizes the covert rate. We further show that this optimal point is determined solely by the network geometry and remains invariant to the covertness requirement $ϵ$.

Comparison with Existing Uncertainty Mechanisms

To clearly position the proposed framework within the existing literature, we compare the selection-induced structural uncertainty with four representative classes of covert communication mechanisms. The key distinctions are summarized in

Table 1. Comparison of uncertainty mechanisms in covert communication.

		Existing Approaches	Proposed Framework
(i) CSI Uncertainty [9,10]	Secret information	Secret pilots/secret codebook	Not required
	Uncertainty source	Channel estimation error	Combinatorial subset ambiguity
	Willie’s CSI	Imperfect (by design)	Perfect (public pilots)
	Reducibility	Degrades with higher SNR or longer observation	Irreducible; independent of N
	Covertness basis	Information asymmetry via secrecy	Structural asymmetry via selection
(ii) Independent Jammer [11,12,13]	Secret information	Secret coordination/pre-shared keys	Not required
	Activation model	$I_m\sim$ i.i.d. Bernoulli$\left(p\right)$	$\sum I_m=K$ (dependent)
	Active count K	Random (Poisson-Binomial)	Deterministic per realization
	Dependence	Independent across nodes	Sampling w/o replacement
	Covert constraint	Statistical (averaged over channel)	Instantaneous (per realization)
(iii) Exact-K w/Secret Pilots [15,16,17]	Secret information	Secret pilot sequences (required)	Not required (public pilots)
	Activation model	$\sum I_m=K$ (same structure)	$\sum I_m=K$ (same structure)
	Willie’s knowledge	${\mathsf{\Gamma }}_m$ unknown (secret pilots)	${\mathsf{\Gamma }}_m$ fully known
	Covertness origin	Hidden jammer-to-Willie channels	Combinatorial ambiguity of $\left({\displaystyle \genfrac{}{}{0pt}{}{M}{K}}\right)$ subsets
	If secret is leaked	Covertness collapses	Unaffected (no secret to leak)
(iv) Stochastic Geometry [14]	Secret information	Secret codebook/hidden interferer identity	Not required
	What is unknown	Where interferers are located	Which K-subset is active
	Network topology	Random (spatial PPP)	Fixed and known to Willie
	Randomness type	Spatial (continuous)	Combinatorial (discrete)
	Analytical tool	Campbell’s theorem/standard CLT	Hájek–Serfling CLT (finite pop.)

.

In CSI uncertainty-based schemes [9,10], Willie’s detection capability improves monotonically with his estimation accuracy, which means covertness erodes as Willie acquires better channel knowledge. In contrast, the proposed framework assumes perfect individual CSI at Willie and demonstrates that covertness persists due to the irreducible combinatorial uncertainty of the active subset $\mathcal{S}$.

In independent jammer models [11,12,13], each jammer independently decides whether to transmit (e.g., Bernoulli activation with probability p), and the aggregate interference follows a sum of independent random variables. In the proposed model, the constraint $\sum I_m=K$ induces a strict dependence structure that cannot be decomposed into independent components. This sampling-without-replacement structure is the mathematical origin of the finite-population correction factor ${\displaystyle \frac{M-K}{M-1}}$ appearing in the selection-induced variance ${\sigma }_X^2$.

It is particularly important to contrast the proposed framework with the prior works [15,16,17], which share the same deterministic exact-K activation structure. The critical distinction is that [15,16,17] rely on secret pilot sequences to prevent Willie from estimating individual interference powers $\left\{{\mathsf{\Gamma }}_m\right\}$; if these secret pilots are compromised, the covertness guarantee collapses entirely. In contrast, the present work operates with public pilots, granting Willie perfect knowledge of every ${\mathsf{\Gamma }}_m$. Covertness arises solely from the $\left({\displaystyle \genfrac{}{}{0pt}{}{M}{K}}\right)$ combinatorial ambiguity of the active subset, a guarantee that is structurally robust because there is no secret information whose leakage could degrade it.

In stochastic geometry frameworks [14], Willie is uncertain about where interferers are located (spatial randomness), whereas in the proposed model, Willie knows the exact locations and channel gains of all M users but remains uncertain about which K-subset is active (combinatorial randomness). The uncertainty source is fundamentally different in nature, i.e., spatial versus combinatorial, and accordingly requires different analytical tools (Campbell’s theorem vs. Hájek–Serfling CLT).

2. System Model and Problem Formulation

In this section, we establish a mathematical framework for a multi-user cooperative network. We consider an uplink scenario where a covert user, Alice, seeks to transmit a message to a base station (BS), Bob, while a set of M cooperative users (CUs) assists the transmission. The system operates under a public pilot-aided protocol, representing a worst-case scenario in which Willie perfectly estimates the instantaneous channel gains of all users but remains uncertain about the active subset S. This subset uncertainty constitutes the structural uncertainty exploited to achieve covertness.

2.1. Network Topology and Spatial Distribution

The network consists of Alice (A), Bob (B), Willie (W), and M cooperative users, $\mathcal{U}=\{U_1,\dots ,U_M\}$. All entities are equipped with a single antenna. We assume frequency division multiple access (FDMA), where Alice is allocated frequency band $f_a$, while each $U_m$ is assigned $f_m$. A subset of users in $\mathcal{U}$ is opportunistically selected to emit interference signals over $f_a$.

Unlike prior models that assume homogeneous user configurations [15,16,17], we explicitly consider spatially dispersed users. Let $d_{ij}$ denote the Euclidean distance between nodes i and j, where $i,j\in \{a,b,w,\mathcal{U}\}$. The large-scale path loss is modeled as $L_{i,j}={\beta }_0{d}_{ij}^{-\alpha }$, where ${\beta }_0$ is the reference gain, and $\alpha$ is the path-loss exponent. Spatial dispersion ensures that $L_{m,w}\ne L_{n,w}$ for $m\ne n$, providing the diversity required to generate selection uncertainty.

2.2. Covertness Metric: Detection Error Probability (DEP)

The objective of covert communication is to prevent Willie from reliably distinguishing between the null hypothesis $H_0$ (no covert transmission) and the alternative hypothesis $H_1$ (covert transmission present). Accordingly, Willie’s detection task is formulated as a binary hypothesis test based on his received signal observations [18].

The performance of Willie’s detector is characterized by two types of errors: the false alarm probability $P_{FA}$, defined as the probability of deciding $H_1$ under $H_0$, and the missed detection probability $P_{MD}$, defined as the probability of deciding $H_0$ under $H_1$. For a given decision threshold $\tau$, the total detection error probability (DEP) is defined as

$$\xi \left(\tau \right)=P_{FA}\left(\tau \right)+P_{MD}\left(\tau \right),$$

where $\tau$ is the detection threshold. A perfectly covert system corresponds to the case where Willie’s detector performs no better than random guessing, yielding $\xi =1$. In practical systems, covertness is quantified through the $ϵ$-covertness criterion, which requires that the minimum achievable DEP satisfies

$$\underset{\tau }{min},\xi \left(\tau \right)\ge 1-ϵ,$$

for a given $ϵ>0$. Therefore, the design objective of the proposed framework is to optimize Alice’s transmission parameters while ensuring that the above covertness constraint is satisfied.

2.3. Proposed Communication Protocol

We consider a structured communication protocol composed of four phases under a public pilot-aided environment. The protocol is designed for a worst-case scenario in which Willie possesses individual channel state information (CSI) and knowledge of transmit power parameters. Despite this strong adversarial assumption, the legitimate system can determine transmission parameters that satisfy the covert constraint through coordinated channel estimation and user selection.

2.3.1. Structure of Channel Notation

To describe the channel estimation process in a unified manner, we introduce the channel notation $h_{y,z}^x$, where the superscript and subscripts explicitly indicate the frequency band and link direction, respectively.

• Superscript (x): The superscript denotes the frequency band over which the channel is estimated. For example, $x=a$ and $x=m$ correspond to the frequency bands $f_a$ and $f_m$, respectively.
• Subscript ($y,z$): The subscripts indicate the directional link associated with the pilot transmission. The first index y represents the transmitting node, while the second index z denotes the receiving node that estimates the channel.
• Examples: $h_{b,m}^m$ denotes the channel from Bob to user m estimated over band $f_m$. Similarly, $h_{a,b}^a$ and $h_{m,b}^a$ represent the uplink channels from Alice and user m to Bob, respectively, measured over Alice’s frequency band $f_a$.

Channel estimation is performed through a two-way pilot exchange within a single coherence interval. Under the channel reciprocity assumption, the estimated channels satisfy

$$h_{y,z}^x=h_{z,y}^x,$$

for all relevant node pairs. This notation enables a consistent description of user selection and power optimization throughout the proposed protocol.

2.3.2. Phase 1: Two-Way Channel Estimation

Channel estimation is performed through a two-way pilot exchange.

• Downlink estimation: Bob broadcasts pilot signals over all frequency bands $\{f_a,f_1,\dots ,f_M\}$. Alice and each cooperative user estimate their respective downlink channels, e.g., $h_{b,a}^a$ and $h_{b,m}^m$. Since pilots are public, Willie also estimates his corresponding channels $h_{b,w}^a$ and $h_{b,w}^m$.
• Uplink estimation: Each user transmits pilot signals back to Bob. Alice transmits over $f_a$, while each $U_m$ transmits over both $f_a$ (for cooperation) and $f_m$ (for its own communication). Bob estimates the uplink channels $h_{a,b}^a$, ${\left\{h_{m,b}^m\right\}}_{m=1}^M$, and ${\left\{h_{m,b}^a\right\}}_{m=1}^M$.
• Adversarial knowledge: Under the public pilot assumption, Willie also observes these uplink pilots and estimates $h_{a,w}^a$ and $h_{m,w}^a$. Therefore, Willie is assumed to possess the full CSI of his own links.

2.3.3. Phase 2: Centralized Parameter Setting at Bob

Based on the estimated uplink CSI from all cooperative users, Bob determines the transmission parameters required to satisfy the covertness constraint. Specifically, Bob computes the transmit power $P_a$ and the activation parameter $\gamma$, which governs user participation in the cooperative set. These parameters are selected such that the resulting system satisfies the covertness requirement

$${\xi }^{*}\ge 1-ϵ.$$

2.3.4. Phase 3: Parameter Broadcast and Adversarial Awareness

Bob broadcasts the parameter set $\{P_a,\gamma \}$ to Alice and all cooperative users. Alice adopts the designated transmit power $P_a$, while each user $U_m$ locally determines its participation in the active subset S according to the activation rule defined by $\gamma$. We assume a worst-case scenario where Willie also intercepts this broadcast, implying that both the transmit power and activation rule are known to the adversary.

2.3.5. Phase 4: Covert Data Transmission and Detection

During data transmission, Alice sends the covert signal $x_a\left[t\right]$ with power $P_a$. Simultaneously, users satisfying the selection criterion participate in the active subset S and transmit masking signals with power $P_c$ over frequency band $f_a$. Willie observes the aggregated received signal and performs binary hypothesis testing. As shown in Section 3, despite having full CSI and protocol knowledge, Willie’s optimal detector reduces to energy detection due to the subset uncertainty associated with the unknown active set S.

2.4. Selection Uncertainty as a Covert Mechanism

The core mechanism of the proposed scheme lies in the decoupling of selection information. While the active subset S is deterministically selected by the legitimate system, it appears stochastic to Willie due to the independence between legitimate and adversarial channels. Alice and Bob select users based on the metric as follows:

$$g_{m,b}={\left|h_{m,b}^a\right|}^2.$$

Without loss of generality, we sort the selection metrics in ascending order as

$$g_{\left(1\right)}\le g_{\left(2\right)}\le \cdots \le g_{\left(M\right)},$$

(1)

where $g_{\left(m\right)}$ denotes the m-th order statistic. Bob sets the activation parameter as

$$\gamma =g_{\left(K\right)},$$

(2)

such that exactly K users are activated. Accordingly, the active subset is defined as

$$S=\{m\mid g_{m,b}\le \gamma \}.$$

(3)

Because the user selection is performed based on the legitimate links, $\left\{h_{m,b}^a\right\}$, which are statistically independent of Willie’s links, $\left\{h_{m,w}\right\}$, each subset $S\in \mathcal{S}$ is equally likely from Willie’s perspective, yielding

$$Pr\left(S\right)={\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{M}{K}\right)}}.$$

(4)

2.5. Signal Model and Hypothesis Models

We consider a quasi-static Rayleigh fading environment where the channel coefficients satisfy

$$h_{i,j}^x\sim \mathcal{CN}(0,L_{i,j}),$$

and remain constant over a block of N symbols. Alice transmits the signal $x_a\left[t\right]$ with transmit power $P_a$ over frequency band $f_a$. Simultaneously, the active subset $S\subset \mathcal{U}$ with cardinality $\left|S\right|=K$ transmits cooperative signals $x_m\left[t\right]$ with power $P_c$. All transmitted symbols are modeled as independent CSCG random variables with unit variance. The received signal at Bob is given by

$$\begin{array}{c}\hfill y\left[t\right]=\left\{\begin{array}{cc}\sum _{m\in S}\sqrt{P_c}{h}_{m,b}^a{x}_m\left[t\right]+n_b\left[t\right],\hfill & H_0,\hfill \\ \sqrt{P_a}{h}_{a,b}^a{x}_a\left[t\right]+\sum _{m\in S}\sqrt{P_c}{h}_{m,b}^a{x}_m\left[t\right]+n_b\left[t\right],\hfill & H_1,\hfill \end{array}\right.\end{array}$$

(5)

where $n_b\left[t\right]\sim \mathcal{CN}(0,{\sigma }_b^2)$. The received signal at Willie is

$$\begin{array}{c}\hfill z\left[t\right]=\left\{\begin{array}{cc}\sum _{m\in S}\sqrt{P_c}{h}_{m,w}^a{x}_m\left[t\right]+n_w\left[t\right],\hfill & H_0,\hfill \\ \sqrt{P_a}{h}_{a,w}^a{x}_a\left[t\right]+\sum _{m\in S}\sqrt{P_c}{h}_{m,w}^a{x}_m\left[t\right]+n_w\left[t\right],\hfill & H_1,\hfill \end{array}\right.\end{array}$$

(6)

where $n_w\left[t\right]\sim \mathcal{CN}(0,{\sigma }_w^2)$.

Although Willie knows ${\left\{h_{m,w}^a\right\}}_{m=1}^M$, he remains uncertain about the active subset S, which constitutes the primary covert mechanism. Since $x_a\left[t\right]$, $x_m\left[t\right]$, and $n_w\left[t\right]$ are independent CSCG random variables, the received signal $z\left[t\right]$ conditioned on S is also CSCG with zero mean,

$$\mathbb{E}[z\left[t\right]\mid S,H_{\theta }]=0.$$

The conditional variance under hypothesis $H_{\theta }$ is

$${\sigma }_z^2(S,H_{\theta })=\theta P_a|h_{a,w}^a{|}^2+\sum _{m\in S}{P}_c{\left|h_{m,w}^a\right|}^2+{\sigma }_w^2,$$

(7)

where $\theta \in \{0,1\}$ corresponds to $H_0$ and $H_1$. The conditional PDF of a single observation is

$$f(z\left[t\right]\mid S,H_{\theta })={\displaystyle \frac{1}{\pi {\sigma }_z^2(S,\theta )}}exp\left(-{\displaystyle \frac{{\left|z\left[t\right]\right|}^2}{{\sigma }_z^2(S,\theta )}}\right),$$

(8)

and the joint conditional PDF over N symbols becomes

$$f(\mathbf{z}\mid S,H_{\theta })={\displaystyle \frac{1}{{\pi }^N{\left({\sigma }_z^2(S,\theta )\right)}^N}}exp\left(-{\displaystyle \frac{{\sum }_{t=1}^N{\left|z\left[t\right]\right|}^2}{{\sigma }_z^2(S,\theta )}}\right),$$

(9)

where $\mathbf{z}={[z\left[1\right],\dots ,z\left[N\right]]}^T$. Applying the law of total probability with the uniform prior over subsets yields the marginal PDF

$$f\left(\mathbf{z}\right|H_{\theta })={\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{M}{K}\right)}}\sum _{S\in \mathcal{S}}{\displaystyle \frac{1}{{\pi }^N{\left({\sigma }_z^2(S,\theta )\right)}^N}}exp\left(-{\displaystyle \frac{{\sum }_{t=1}^N{\left|z\left[t\right]\right|}^2}{{\sigma }_z^2(S,\theta )}}\right),$$

(10)

which forms a Gaussian mixture model. This mixture structure prevents Willie from identifying the active subset even with full CSI knowledge. Here, $H_0$ denotes the absence of Alice’s covert transmission, while cooperative users may still transmit masking signals.

3. Mathematical Optimality of Energy Detection

In this section, we provide a rigorous proof that an energy-based detector remains optimal for Willie, even when he possesses full CSI of all potential links. We demonstrate that the selection-based uncertainty neutralizes the advantage of individual CSI, and the optimal test reduces to a function of the average received power.

3.1. Likelihood Ratio Test (LRT) and Sufficient Statistic

According to the Neyman–Pearson lemma [18,19], the optimal decision rule that minimizes the total detection error probability (DEP) is the Likelihood Ratio Test (LRT). Let $\mathbf{z}={[z\left[1\right],\dots ,z\left[N\right]]}^T$ denote Willie’s observation vector. We define the average received power as the test statistic

$$T_w\triangleq {\displaystyle \frac{1}{N}}\sum _{t=1}^N{\left|z\left[t\right]\right|}^2.$$

(11)

Using the joint conditional PDF in (9), the marginal likelihood under hypothesis $H_{\theta }$ is obtained by marginalizing over all subsets $S\in \mathcal{S}$ as

$$f(\mathbf{z}\mid H_{\theta })={\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{M}{K}\right)}}\sum _{S\in \mathcal{S}}f(\mathbf{z}\mid S,H_{\theta }).$$

(12)

Substituting (9) and (12) into the LRT yields

$$\mathsf{\Lambda }\left(\mathbf{z}\right)\triangleq {\displaystyle \frac{f(\mathbf{z}\mid H_1)}{f(\mathbf{z}\mid H_0)}}={\displaystyle \frac{{\sum }_{S\in \mathcal{S}}{\left({\sigma }_z^2(S,1)\right)}^{-N}exp\left(-\frac{N{T}_w}{{\sigma }_z^2(S,1)}\right)}{{\sum }_{S\in \mathcal{S}}{\left({\sigma }_z^2(S,0)\right)}^{-N}exp\left(-\frac{N{T}_w}{{\sigma }_z^2(S,0)}\right)}}\underset{H_0}{\overset{H_1}{\gtrless }}\lambda ,$$

(13)

where $\lambda$ denotes the decision threshold. The common scaling constants $1/\left({\displaystyle \genfrac{}{}{0pt}{}{M}{K}}\right)$ and $1/{\pi }^N$ cancel out in the likelihood ratio.

3.2. Sufficient Statistic and Fisher–Neyman Factorization

We now identify the optimal detection statistic using the Fisher–Neyman factorization theorem [19].

Theorem 1. 

In the presence of selection uncertainty, the average received power

$$T_w={\displaystyle \frac{1}{N}}\sum _{t=1}^N{\left|z\left[t\right]\right|}^2$$

is a sufficient statistic for Willie’s binary hypothesis test.

Proof. 

From (12), the marginal PDF under hypothesis $H_{\theta }$ can be written as

$$\begin{array}{cc}\hfill f(\mathbf{z}\mid H_{\theta })& ={\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{M}{K}\right)}}\sum _{S\in \mathcal{S}}{\displaystyle \frac{1}{{\pi }^N{\left({\sigma }_z^2(S,\theta )\right)}^N}}exp\left(-{\displaystyle \frac{N{T}_w}{{\sigma }_z^2(S,\theta )}}\right)\hfill \\ \hfill & =\underset{h\left(\mathbf{z}\right)}{\underbrace{1}}·\underset{g(T_w,H_{\theta })}{\underbrace{{\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{M}{K}\right){\pi }^N}}{\sum }_{S\in \mathcal{S}}{\left({\sigma }_z^2(S,\theta )\right)}^{-N}exp\left(-{\displaystyle \frac{N{T}_w}{{\sigma }_z^2(S,\theta )}}\right)}}.\hfill \end{array}$$

(14)

Here, $h\left(\mathbf{z}\right)$ is independent of the hypothesis $H_{\theta }$, while $g(T_w,H_{\theta })$ depends on $\mathbf{z}$ only through the scalar statistic $T_w$. By the Fisher–Neyman factorization theorem, $T_w$ is therefore a sufficient statistic. Consequently, the likelihood ratio can be expressed as

$$\mathsf{\Lambda }\left(\mathbf{z}\right)=\varphi \left(T_w\right),$$

for some function $\varphi (·)$, which completes the proof. □

3.3. Monotonicity and Optimal Decision Rule

To implement the LRT in practice, consider the log-likelihood ratio (LLR),

$$L\left(T_w\right)\triangleq ln\varphi \left(T_w\right),$$

which can be written as

$$L\left(T_w\right)=ln\left(\sum _{S\in \mathcal{S}}{\alpha }_S^{\left(1\right)}exp\left(-{\displaystyle \frac{N{T}_w}{{\sigma }_z^2(S,1)}}\right)\right)-ln\left(\sum _{S\in \mathcal{S}}{\alpha }_S^{\left(0\right)}exp\left(-{\displaystyle \frac{N{T}_w}{{\sigma }_z^2(S,0)}}\right)\right),$$

where

$${\alpha }_S^{\left(\theta \right)}\triangleq {\left({\sigma }_z^2(S,\theta )\right)}^{-N}$$

are positive constants for a given channel realization. Under the covert transmission model, for any fixed subset S, the conditional variances satisfy

$${\sigma }_z^2(S,1)={\sigma }_z^2(S,0)+P_a{\left|h_{a,w}^a\right|}^2>{\sigma }_z^2(S,0),$$

(15)

since $P_a>0$. Consequently, the likelihood ratio becomes a monotone function of the sufficient statistic $T_w$. Therefore, the LRT can be equivalently expressed as a threshold test on the received energy,

$$T_w={\displaystyle \frac{1}{N}}\sum _{t=1}^N{\left|z\left[t\right]\right|}^2\underset{H_0}{\overset{H_1}{\gtrless }}\tau ,$$

(16)

where $\tau$ denotes the detection threshold. This result shows that, even when Willie possesses full CSI, the uncertainty associated with the unknown active subset S leads to an energy-based optimal decision rule.

4. Covertness Analysis Without Secret Pilot

In this section, we provide a rigorous statistical characterization of Willie’s detection performance. The proposed framework removes the reliance on pre-shared secret pilots by exploiting the structural uncertainty inherent in multi-user networks. To establish a conservative security benchmark, we consider a worst-case scenario in which Willie possesses an infinite observation window, $N\to \infty$, and full knowledge of the individual channel gains, ${\left\{h_{m,w}^a\right\}}_{m=1}^M$ and $h_{a,w}^a$, obtained through public pilots. Under this asymptotic regime, temporal randomness caused by symbol-level fluctuations and thermal noise vanishes, and covertness relies solely on the structural uncertainty induced by the unknown active subset S.

4.1. Mathematical Derivation of the Asymptotic Test Statistic

To analyze Willie’s detection capability, we investigate the asymptotic behavior of the average received power $T_w$. For a given active subset $S\in \mathcal{S}$ and hypothesis $H_{\theta }$, Willie’s received signal at time index $t\in \{1,\dots ,N\}$ is modeled as

$$z\left[t\right]=\theta \sqrt{P_a}{h}_{a,w}^a{x}_a\left[t\right]+\sum _{m\in S}\sqrt{P_c}{h}_{m,w}^a{x}_m\left[t\right]+n_w\left[t\right],$$

(17)

where $x_a\left[t\right]$ and $x_m\left[t\right]$ are independent CSCG symbols with unit variance, and $n_w\left[t\right]\sim \mathcal{CN}(0,{\sigma }_w^2)$ denotes AWGN. Within one coherence interval, the channel coefficients $h_{a,w}^a$ and $h_{m,w}^a$ are assumed quasi-static and are treated as deterministic quantities known to Willie. Conditioned on $(S,H_{\theta })$, the observations ${\left\{z\left[t\right]\right\}}_{t=1}^N$ are independent and identically distributed CSCG random variables,

$$z\left[t\right]\mid S,H_{\theta }\sim \mathcal{CN}(0,{\sigma }_z^2(S,\theta )),$$

(18)

where the conditional variance is given by

$${\sigma }_z^2(S,\theta )={\mathbb{E}\left[\right|z\left[t\right]|}^2\mid S,H_{\theta }]=\theta P_a|h_{a,w}^a{|}^2+\sum _{m\in S}{P}_c{\left|h_{m,w}^a\right|}^2+{\sigma }_w^2.$$

(19)

Since ${\left|z\left[t\right]\right|}^2$ follows a scaled chi-square distribution with two degrees of freedom, i.e.,

$${\left|z\left[t\right]\right|}^2\sim {\displaystyle \frac{{\sigma }_z^2(S,\theta )}{2}}{\chi }_2^2,$$

and the Strong Law of Large Numbers (SLLN) [20] yields

$$\mathbb{P}\left(\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\sum _{t=1}^N{\left|z\left[t\right]\right|}^2={\mathbb{E}\left[\right|z\left[t\right]|}^2\mid S,H_{\theta }]\right)=1.$$

(20)

Defining

$$\mathsf{\Delta }\triangleq P_a|h_{a,w}^a{|}^2,{\mathsf{\Gamma }}_m\triangleq P_c{\left|h_{m,w}^a\right|}^2,$$

the asymptotic limit of the test statistic becomes

$$T_w=\theta \mathsf{\Delta }+\sum _{m\in S}{\mathsf{\Gamma }}_m+{\sigma }_w^2,N\to \infty .$$

(21)

Equation (21) shows that, with an infinite observation window, temporal randomness vanishes and Willie’s observation converges to a deterministic sum of power components for a fixed subset S. However, because the subset selection process is independent of Willie’s channel knowledge, the combinatorial uncertainty of S remains, constituting the fundamental source of covertness.

4.2. Statistical Modeling of Selection Uncertainty

In the proposed framework, the information asymmetry required for covertness does not originate from pre-shared secret sequences but from the combinatorial uncertainty induced by multi-user selection. Under the public pilot assumption, the individual received power components, ${\mathsf{\Gamma }}_m$, and the covert signal power, $\mathsf{\Delta }$, are deterministic quantities known to Willie. However, due to the statistical independence between the legitimate links and Willie’s channels, the mapping between these known power components and the active subset S remains unknown. As a result, Willie’s detection problem becomes a stochastic inference problem over a discrete combinatorial space.

4.2.1. Small-K Regime: Combinatorial Ambiguity and the Subset-Sum Structure

When the number of active cooperative users, K, is small, the asymptotic test statistic $T_w$ as $N\to \infty$ takes values on a finite discrete support. Let ${\mathcal{Y}}_{\theta }$ denote the set of all possible aggregate power levels under hypothesis $H_{\theta }$, as follows:

$${\mathcal{Y}}_{\theta }=\left\{y|y=\sum _{m\in S}{\mathsf{\Gamma }}_m+\theta \mathsf{\Delta }+{\sigma }_w^2,S\in \mathcal{S}\right\}.$$

(22)

The cardinality of this support is

$$|{\mathcal{Y}}_{\theta }|=\left({\displaystyle \genfrac{}{}{0pt}{}{M}{K}}\right).$$

From Willie’s perspective, the observation $T_w$ follows a discrete distribution given by

$$P(T_w=y\mid H_{\theta })={\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{M}{K}\right)}}\sum _{S\in \mathcal{S}}{\delta }_K\left(y-\left(\sum _{m\in S}{\mathsf{\Gamma }}_m+\theta \mathsf{\Delta }+{\sigma }_w^2\right)\right),$$

(23)

where ${\delta }_K(·)$ denotes the Kronecker delta function.

In this regime, the detection problem can be interpreted as identifying whether the additional power component $\mathsf{\Delta }$ is present within a combinatorial subset-sum structure. When the spatial dispersion among users is sufficiently large, the power levels in ${\mathcal{Y}}_0$ become widely distributed, and the shifted set ${\mathcal{Y}}_1$ densely interleaves with ${\mathcal{Y}}_0$. Consequently, a single energy observation may correspond to either hypothesis, which preserves covertness even without pre-shared secret pilots.

4.2.2. Large-K Regime: Gaussian Approximation of the Test Statistic

While the small-K regime is characterized by discrete combinatorial ambiguity, the detection problem transitions into a continuous statistical regime as the number of active cooperative users K increases. As the discrete power levels described in Section 4.2.1 become dense, Willie’s detection problem shifts from a discrete search to distinguishing distributions with continuous support.

Our objective in this subsection is to derive an approximate probability density function (PDF) of the test statistic $T_w$ in the large-K regime. Recall that, conditioned on a fixed active subset S and hypothesis $H_{\theta }$, the Strong Law of Large Numbers yields

$$T_w\stackrel{\mathrm{a}.\mathrm{s}.}{\to }{\sigma }_z^2(S,\theta )=\theta \mathsf{\Delta }+\sum _{m\in S}{\mathsf{\Gamma }}_m+{\sigma }_w^2,$$

(24)

as $N\to \infty$. Hence, in the asymptotic observation regime, the randomness of $T_w$ originates solely from the unknown active subset S. Define the aggregate interference term as

$$X\triangleq \sum _{m\in S}{\mathsf{\Gamma }}_m.$$

(25)

Since S is uniformly selected from all K-subsets of $\{1,\dots ,M\}$, we equivalently write

$$X=\sum _{m=1}^M{I}_m{\mathsf{\Gamma }}_m,$$

(26)

where $I_m\in \{0,1\}$ are dependent indicator variables satisfying ${\sum }_{m=1}^M{I}_m=K$. Therefore, the distribution of $T_w$ is fully determined by the distribution of X. By symmetry of uniform sampling without replacement, $\mathbb{E}\left[I_m\right]=K/M$, and the mean of X is

$${\mu }_X\triangleq \mathbb{E}\left[X\right]={\displaystyle \frac{K}{M}}\sum _{m=1}^M{\mathsf{\Gamma }}_m.$$

(27)

Accounting for dependence among $\left\{I_m\right\}$, the variance of X is

$${\sigma }_X^2\triangleq \mathrm{Var}\left(X\right)={\displaystyle \frac{K(M-K)}{M-1}}{\sigma }_{\mathsf{\Gamma }}^2,$$

(28)

where

$${\sigma }_{\mathsf{\Gamma }}^2={\displaystyle \frac{1}{M}}\sum _{m=1}^M{\mathsf{\Gamma }}_m^2-{\left({\displaystyle \frac{1}{M}}\sum _{m=1}^M{\mathsf{\Gamma }}_m\right)}^2$$

(29)

denotes the population variance of the interference powers. This variance captures the structural uncertainty induced by subset selection and reflects the spatial dispersion among cooperative users.

We now approximate the distribution of X. When $K\gg 1$, and the population size satisfies $M\gg K$, the aggregate interference consists of many non-dominant components. Under this condition, a central limit theorem for finite-population sampling, such as the Hájek–Serfling CLT [21], can be applied. One sufficient condition for Gaussian convergence is the Lyapunov-type criterion

$$\underset{M\to \infty }{lim}{L}_M=\underset{M\to \infty }{lim}{\displaystyle \frac{{\sum }_{m=1}^M{|{\mathsf{\Gamma }}_m-\overline{\mathsf{\Gamma }}|}^3}{{\left(M{\sigma }_{\mathsf{\Gamma }}^2\right)}^{3/2}}}=0,\overline{\mathsf{\Gamma }}={\displaystyle \frac{1}{M}}\sum _{m=1}^M{\mathsf{\Gamma }}_m,$$

(30)

which excludes the presence of a dominant interferer. Under this condition, the aggregate interference can be approximated as

$$X\sim \mathcal{N}({\mu }_X,{\sigma }_X^2).$$

(31)

Consequently, the asymptotic distribution of the test statistic under hypothesis $H_{\theta }$ becomes

$$T_w\sim \mathcal{N}\left(\theta \mathsf{\Delta }+{\mu }_X+{\sigma }_w^2,{\sigma }_X^2\right).$$

(32)

Therefore, although temporal randomness vanishes as $N\to \infty$, the selection-induced variance remains finite and constitutes an irreducible structural uncertainty in Willie’s observation. As a result, the covert power shift $\mathsf{\Delta }$ is statistically masked by the Gaussian uncertainty generated by subset selection.

4.2.3. Robustness Against an Ideal Willie

The proposed selection uncertainty provides a more robust covertness mechanism than conventional secret-pilot-based schemes. In secret pilot models, covertness relies on preventing Willie from accurately estimating the channel, i.e., CSI uncertainty. However, a Willie operating at high SNR or equipped with advanced estimation techniques can progressively reduce this uncertainty. In contrast, the proposed framework exploits structural uncertainty, which persists even when Willie possesses perfect knowledge of individual channel realizations. The selection-induced variance ${\sigma }_X^2$ originates from the random subset formation process and therefore cannot be eliminated through longer observation windows or improved channel estimation. Specifically, ${\sigma }_X^2$ depends on the spatial dispersion of the path-loss components $\left\{L_{m,w}\right\}$ as well as the network scale parameters M and K. Since this variance is independent of the observation length N, it remains as an irreducible uncertainty term in Willie’s detection process. This indicates that a positive covert rate can be sustained even against a Willie with perfect CSI, provided that the legitimate system maintains a sufficiently diverse pool of cooperative users whose channels to Willie are statistically independent of their channels to the base station.

Remark 1 

(Distinction from Classical Interference Randomness). The selection-induced uncertainty in the proposed framework is fundamentally distinct from existing stochastic interference models for covert communication. Single-jammer models [11,12] create Willie’s uncertainty through random power variation of a single node, which is a one-dimensional uncertainty mechanism structurally different from combinatorial subset selection. In the multi-user cooperative jamming model of [13], each jammer activates independently with non-identical probabilities, making K a random variable (Poisson-Binomial); the covert constraint is satisfied statistically in expectation. In contrast, our model enforces a deterministic exact-K selection per channel realization, restricting the activation vector to the Hamming sphere ${\mathcal{H}}_K$ and inducing negative dependence ($\mathrm{Pr}(I_n=1\mid I_m=1)={\displaystyle \frac{K-1}{M-1}}\ne {\displaystyle \frac{K}{M}}$). Even under the worst-case assumption that Willie knows K, the $\left({\displaystyle \genfrac{}{}{0pt}{}{M}{K}}\right)$ combinatorial ambiguity guarantees covertness per realization. While our prior works [15,16,17] employ the same deterministic exact-K activation structure, they rely on secret pilot sequences to hide the user-to-Bob channels from Willie. The present work eliminates this requirement: covertness is achieved solely through the combinatorial ambiguity of the active subset under public pilots, with Willie possessing perfect knowledge of all individual interference powers $\left\{{\mathsf{\Gamma }}_m\right\}$. In stochastic geometry frameworks [14], uncertainty arises from random spatial locations, whereas in the proposed model, the network topology is fully known to Willie, and the uncertainty is purely combinatorial. The aggregate interference variance ${\sigma }_X^2={\displaystyle \frac{K(M-K)}{M-1}}{\sigma }_{\mathsf{\Gamma }}^2$ contains a finite-population correction arising from exact-K sampling without replacement [21], a structure absent in all independent-activation models. Consequently, the proposed model cannot be expressed as a special case of existing stochastic interference frameworks.

5. Detection Error Probability

In this section, we characterize the detection error probability (DEP) to evaluate the covertness performance of the proposed framework. We focus on the worst-case scenario in which Willie employs the optimal energy detector derived in Section 3. In the asymptotic regime ($N\to \infty$), the detection performance is limited solely by the structural uncertainty analyzed in Section 4.

5.1. General DEP Formulation

Willie’s objective is to minimize the total detection error probability, defined as the sum of the false alarm probability and the missed detection probability as follows:

$$\xi =P_{\mathrm{FA}}+P_{\mathrm{MD}}=\mathbb{P}(T_w>\tau \mid H_0)+\mathbb{P}(T_w\le \tau \mid H_1).$$

(33)

In the asymptotic regime, the randomness of $T_w$ originates from the subset-selection uncertainty, which induces the discrete support ${\mathcal{Y}}_{\theta }$ defined in Section 4.2. Under the uniform prior assumption over all feasible subsets, the DEP for a given threshold $\tau$ can be expressed as

$$\xi \left(\tau \right)={\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{M}{K}\right)}}\sum _{y\in {\mathcal{Y}}_0}\mathbb{I}(y>\tau )+{\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{M}{K}\right)}}\sum _{y\in {\mathcal{Y}}_1}\mathbb{I}(y\le \tau ),$$

(34)

where $\mathbb{I}(·)$ denotes the indicator function.

This formulation shows that Willie’s detection performance is governed by the degree of overlap between the two discrete supports ${\mathcal{Y}}_0$ and ${\mathcal{Y}}_1$.

5.2. Detection Performance in the Small-K Regime

As discussed in Section 4.2.1, the support ${\mathcal{Y}}_{\theta }$ remains sparse and discrete when K is small. In this regime, the detection problem reduces to distinguishing two shifted collections of subset sums. The resulting DEP depends on the relative spacing between the elements of ${\mathcal{Y}}_0$ and ${\mathcal{Y}}_1$. When the spatial dispersion among cooperative users is sufficiently large, the resulting aggregate power levels become widely scattered. Consequently, multiple realizations of the active subset under $H_0$ may overlap with those under $H_1$, making the two hypotheses difficult to distinguish and maintaining a high DEP regardless of the observation length N.

5.3. Detection Performance in the Large-K Regime

In the large-K regime, we employ the Gaussian approximation derived in Section 4.2.2, where the test statistic follows

$$T_w\sim \mathcal{N}({\mu }_{\theta },{\sigma }_X^2),$$

with

$${\mu }_0={\mu }_X+{\sigma }_w^2,{\mu }_1={\mu }_X+\mathsf{\Delta }+{\sigma }_w^2.$$

5.3.1. False Alarm and Missed Detection Probabilities

For a fixed detection threshold $\tau$, the false alarm probability and missed detection probability are expressed using the Q-function as

$$P_{\mathrm{FA}}\left(\tau \right)=Q\left({\displaystyle \frac{\tau -({\mu }_X+{\sigma }_w^2)}{{\sigma }_X}}\right),$$

(35)

and

$$P_{\mathrm{MD}}\left(\tau \right)=1-Q\left({\displaystyle \frac{\tau -({\mu }_X+\mathsf{\Delta }+{\sigma }_w^2)}{{\sigma }_X}}\right).$$

(36)

The variance ${\sigma }_X^2$, induced by subset selection, acts as an irreducible structural uncertainty that limits Willie’s detection capability.

5.3.2. Optimal Threshold and Minimum DEP

Since the two Gaussian distributions share the same variance ${\sigma }_X^2$, the threshold minimizing the total DEP is given by

$${\tau }^{*}={\mu }_X+{\displaystyle \frac{\mathsf{\Delta }}{2}}+{\sigma }_w^2.$$

(37)

Substituting ${\tau }^{*}$ into the DEP expression yields

$$\begin{array}{cc}\hfill {\xi }^{*}& =2Q\left({\displaystyle \frac{\mathsf{\Delta }}{2{\sigma }_X}}\right)=2Q\left({\displaystyle \frac{P_a{\left|h_{a,w}^a\right|}^2}{2{\sigma }_X}}\right).\hfill \end{array}$$

(38)

5.4. Covertness Condition and Power Allocation

To satisfy the $ϵ$-covertness requirement, the system must ensure

$${\xi }^{*}\ge 1-ϵ.$$

Using (38), this condition imposes an upper bound on Alice’s transmit power.

Proposition 1. 

In a public pilot-aided environment with selection uncertainty, the transmit power must satisfy

$$P_a\le {\displaystyle \frac{2{\sigma }_X}{|h_{a,w}^a{|}^2}}{Q}^{-1}\left({\displaystyle \frac{1-ϵ}{2}}\right).$$

(39)

The bound in (39) shows that the allowable transmit power scales with the square root of the selection-induced variance ${\sigma }_X^2$. Therefore, as long as spatial dispersion among cooperative users ensures ${\sigma }_X^2>0$, a positive covert transmission power remains feasible even against an ideal Willie.

5.5. Maximum Achievable Covert Rate

Unlike Willie, Bob has perfect knowledge of the active subset S, allowing him to treat the active cooperative interference as deterministic. For a given subset S, the received SINR at Bob is

$${\mathrm{SINR}}_B={\displaystyle \frac{P_a{\left|h_{a,b}^a\right|}^2}{{\sum }_{m\in S}{P}_c{\left|h_{m,b}^a\right|}^2+{\sigma }_b^2}}.$$

(40)

The maximum achievable covert rate is obtained by evaluating the Shannon capacity at the boundary of the covertness constraint. Substituting the maximum allowable power from (39), the covert rate is given by

$$R=W{log}_2\left(1+{\mathrm{SINR}}_B\right),$$

(41)

where W denotes the system bandwidth. This analytical framework provides a tractable characterization of the trade-off between covertness and spectral efficiency. The closed-form expressions enable systematic optimization of the system parameters while satisfying the covertness constraint without relying on pre-shared secret pilots.

6. Numerical Results and Discussion

In this section, we present numerical results to validate the theoretical analysis and to provide insights into the performance of the proposed public pilot-aided covert communication framework. To evaluate the selection uncertainty-based covert mechanism, simulations are conducted under a customized environment following the 3GPP TR 25.996 Urban Macrocell (UMa) model [22]. The simulation setup is designed to represent a conservative worst-case scenario and is described as follows.

6.1. Spatial Topology and Worst-Case Scenario Configuration

The network is deployed over a $1200\times 1200$ ${\mathrm{m}}^2$ area. Willie is located at ${\mathbf{p}}_w=(500,500)$. Alice is placed at ${\mathbf{p}}_a=(831,831)$, yielding an Alice-to-Willie distance of approximately $d_{a,w}\approx 468$ m, and Bob is positioned at ${\mathbf{p}}_b=(100,100)$.

To establish a conservative benchmark for covertness, all $M=1000$ cooperative users are uniformly distributed within an annular region satisfying $d_{m,w}>d_{a,w}$ for all $m\in \{1,\dots ,M\}$. Under this geometry, each CU experiences weaker large-scale channel gain to Willie than Alice, i.e., $L_{m,w}<L_{a,w}$. Consequently, covertness cannot be achieved through strong single-user interference and must instead rely on the combinatorial uncertainty induced by subset selection.

We note that the two key parameters, N (observation window) and M (network size), play physically decoupled roles. The assumption $N\to \infty$ adopted throughout the analysis constitutes a worst-case adversarial model, granting Willie an unlimited observation capability; any finite N would only strengthen the covertness guarantee, since thermal noise and symbol-level fluctuations provide additional masking. In contrast, M is determined by the deployment scenario, and the accuracy of the Gaussian approximation in Section 4.2.2 depends on K and M independently of N. To verify that the Lyapunov condition in (30) is satisfied under the present simulation settings, we numerically evaluate the ratio $L_M$ for $M=1000$ users uniformly distributed in the annular region with $d_{m,w}>d_{a,w}\approx 468$ m. The resulting value is $L_M\approx 0.05$, confirming that no single user dominates the aggregate interference, and the Gaussian approximation is valid well before the asymptotic limit. This is further corroborated by the Monte-Carlo results in Figure 1, where the analytical curve closely matches the simulation even at the moderate value $K=20$ ($2\%$ of the population). While a joint scaling regime $M/N\to c$ does not arise naturally from the current model, since N governs temporal averaging, and M governs combinatorial uncertainty, analyzing their interplay under coupled scaling constitutes an interesting direction for future work.

6.2. Path-Loss Model and Parameter Derivation

The large-scale fading follows the 3GPP Urban Macrocell (UMa) path-loss model, expressed in dB scale as

$$PL\left(d\right)\left[\mathrm{dB}\right]=34.5+35{log}_{10}\left(d\right),$$

(42)

where d denotes the horizontal distance. The 3D distance is approximated by the horizontal distance due to the relatively small height difference in the considered terrestrial scenario. This empirical expression is equivalently written in linear scale using the power-law model,

$$L_{i,j}={\beta }_0{d}_{i,j}^{-\alpha },$$

(43)

where $\alpha$ is the path-loss exponent, and ${\beta }_0$ is the reference channel gain at a distance of $1m$. By matching the logarithmic slope, the path-loss exponent is obtained as

$$10\alpha =35\Rightarrow \alpha =3.5.$$

(44)

Similarly, the reference gain is derived from the intercept term as follows:

$$-10{log}_{10}\left({\beta }_0\right)=34.5\Rightarrow {\beta }_0={10}^{-3.45}\approx 3.55\times {10}^{-4}.$$

(45)

The simulation parameters used throughout this section are summarized in

Table 2. Simulation parameters.

Parameter	Symbol	Value
System Bandwidth	W	5 MHz
Transmit Power of CUs	$P_c$	200 mW (23 dBm)
Noise Power Spectral Density (W/Hz)	${\sigma }_b^2,{\sigma }_w^2$	$-102$ dBm
Path-loss Exponent	$\alpha$	3.5
Reference Channel Gain	${\beta }_0$	$3.55\times {10}^{-4}$
Total Potential Users	M	1000
Active User Count	K	20

.

6.3. Performance Validation and Analysis

Figure 1 illustrates the minimum detection error probability (DEP), ${\xi }^{*}$, at Willie as a function of the target covertness level $ϵ$. The simulation is performed under a worst-case assumption in which Willie possesses perfect knowledge of all individual channel gains $\left\{\right|h_{m,w}^a{{|}^2\}}_{m=1}^M$. The results indicate that covertness is primarily achieved through the structural uncertainty induced by subset selection rather than by additive noise. By selecting K active users from a large candidate pool of size M, the legitimate system introduces combinatorial uncertainty that masks Alice’s transmission. The close agreement between the Monte-Carlo results and the analytical $1-ϵ$ curve confirms the accuracy of the theoretical analysis. In particular, the selection-induced variance,

$${\sigma }_X^2={\displaystyle \frac{K(M-K)}{M-1}}{\sigma }_{\mathsf{\Gamma }}^2,$$

acts as a stable source of structural uncertainty that limits Willie’s detection capability.

Furthermore, the results for $K=100$ exhibit tighter agreement with the theoretical bound compared to the case of $K=20$. This observation is consistent with the Hájek–Serfling central limit theorem, which predicts improved Gaussian approximation accuracy as the number of active users increases. As the aggregate interference becomes statistically more predictable in large-scale networks with spatial dispersion, Alice can accurately determine the optimal transmit power $P_a^{*}$ while satisfying the prescribed covertness constraint. Notably, this behavior is achieved without relying on any secret pilot-based coordination.

Figure 2 shows the achievable covert rate R as a function of the number of cooperative users M. Unlike conventional wireless systems, where increasing the number of nodes typically results in stronger aggregate interference, the proposed framework exhibits an increasing covert rate as M grows. This behavior can be explained by what may be termed selection diversity gain. As the candidate pool size increases, Bob obtains a higher probability of selecting K active cooperative users that generate minimal interference toward the legitimate receiver while still contributing to uncertainty at Willie. The effect can be interpreted through two complementary mechanisms as follows:

• Local interference suppression: A larger M enables more selective user activation at Bob. By ordering the selection metric $g_{m,b}$ and choosing the K smallest values, the effective interference power observed at Bob is reduced, resulting in an improved reception condition.
• Preservation of structural uncertainty: Although the interference at Bob is reduced, the selection-induced variance at Willie, ${\sigma }_X^2$, remains significant, because it depends on the statistical dispersion of the entire user population with respect to Willie.

As a result, increasing the network scale improves the trade-off between covertness and communication efficiency. A larger candidate set allows Alice to employ a higher transmit power $P_a^{*}$ while maintaining a favorable reception condition at Bob. These results indicate that, in networks with spatial dispersion, a large user population can serve as a structural resource that enhances covertness without relying on secret pilot-based coordination or cryptographic keys.

Figure 3 investigates the impact of spatial dispersion on the achievable covert rate R. In this setting, the spatial dispersion is quantified by the distance standard deviation ${\sigma }_d$, which characterizes the geographical spread of cooperative users around Willie. Following the Gaussian dispersion model, the distances $d_{m,w}$ are generated according to a truncated normal distribution,

$$d_{m,w}\sim \mathcal{N}({\mu }_d,{\sigma }_d^2),d_{m,w}\ge d_{a,w}+10\mathrm{m}.$$

The large-scale channel gain is given by $L_{m,w}={\beta }_0{d}_{m,w}^{-\alpha }$, while the small-scale fading satisfies

$$h_{m,w}^a\sim \mathcal{CN}(0,L_{m,w}).$$

The interference contribution of each CU is defined as ${\mathsf{\Gamma }}_m=P_c{\left|h_{m,w}^a\right|}^2$. Since Willie observes the received energy over multiple symbols, the effect of small-scale fading is averaged out, yielding

$$\mathbb{E}\left[\right|h_{m,w}^a{|}^2]=L_{m,w}.$$

Consequently, the variance of the interference power, denoted by ${\sigma }_{\mathsf{\Gamma }}^2=\mathrm{Var}\left({\mathsf{\Gamma }}_m\right)$, is primarily determined by the spatial dispersion ${\sigma }_d^2$. Applying a first-order Taylor approximation around the mean distance ${\mu }_d$, the variance is approximated as

$${\sigma }_{\mathsf{\Gamma }}^2\approx {\left({\left.{\displaystyle \frac{\partial \left(P_c{\beta }_0{d}_{m,w}^{-\alpha }\right)}{\partial d_{m,w}}}\right|}_{d_{m,w}={\mu }_d}\right)}^2{\sigma }_d^2={\alpha }^2{P}_c^2{\beta }_0^2{\mu }_d^{-2(\alpha +1)}{\sigma }_d^2.$$

(46)

Equation (46) shows that ${\sigma }_{\mathsf{\Gamma }}^2$ increases quadratically with spatial dispersion. Since the selection-induced variance satisfies

$${\sigma }_X^2={\displaystyle \frac{K(M-K)}{M-1}}{\sigma }_{\mathsf{\Gamma }}^2,$$

larger spatial dispersion directly increases the structural uncertainty observed by Willie. The monotonic increase in R with respect to ${\sigma }_d$ can be interpreted through the following observations:

• Increase in structural uncertainty: Larger ${\sigma }_d$ implies that CUs are more widely distributed around Willie. Due to the nonlinear path-loss relationship, moderate distance variations produce significant fluctuations in received power, thereby increasing ${\sigma }_X^2$.
• Expansion of the covert operating region: As spatial dispersion increases, the aggregate interference becomes more sensitive to subset selection, allowing Alice to employ a larger transmit power $P_a^{*}$ while maintaining the covertness constraint.
• Persistence under strict covertness constraints: Even under stringent requirements, e.g., $ϵ=0.01$, large spatial dispersion enables non-negligible covert rates, indicating that geometry-induced uncertainty remains effective without relying on secret pilot-based coordination.

Overall, Figure 3 demonstrates that spatial dispersion enhances covertness by increasing selection-induced uncertainty. These results indicate that geometric variability, when combined with subset selection, can be effectively exploited as a structural resource for covert communication.

Figure 4 illustrates the achievable covert rate R as a function of the number of active cooperative users K. The objective of this analysis is to identify the optimal subset size $K^{★}$ that maximizes the covert throughput while satisfying the covertness constraint. A key observation from Figure 4 is that the optimal operating point $K^{★}$ remains nearly invariant with respect to the target covertness level $ϵ$. This behavior can be explained by examining the structure of the received SINR at Bob. From the analytical result in Section 5.4, the maximum allowable transmit power under the $ϵ$-covertness constraint can be written as

$$P_a^{★}(K,ϵ)=\underset{\mathcal{G}\left(K\right)}{\underbrace{{\displaystyle \frac{2\sqrt{{\sigma }_X^2\left(K\right)}}{|h_{a,w}^a{|}^2}}}}\underset{\mathcal{C}\left(ϵ\right)}{\underbrace{Q^{-1}\left({\displaystyle \frac{1-ϵ}{2}}\right)}},$$

(47)

where $\mathcal{G}\left(K\right)$ captures the selection-induced structural uncertainty determined by the network geometry, while $\mathcal{C}\left(ϵ\right)$ is a scalar factor determined solely by the target covertness level. The achievable rate is a monotonic function of the received SINR at Bob, which can be expressed as

$${\mathrm{SINR}}_B(K,ϵ)=\mathcal{C}\left(ϵ\right)·{\displaystyle \frac{\mathcal{G}\left(K\right)|h_{a,b}^a{|}^2}{{\sum }_{m\in S}{P}_c{\left|h_{m,b}^a\right|}^2+{\sigma }_b^2}}.$$

(48)

Since $\mathcal{C}\left(ϵ\right)$ acts as a multiplicative scaling constant, it uniformly scales the SINR magnitude without altering its dependence on K. Consequently, the stationary condition $\partial R/\partial K=0$ yields an optimal solution $K^{★}$ that is independent of $ϵ$. This indicates that the optimal user-selection policy remains structurally robust across different covertness requirements, regardless of whether the constraint is stringent (e.g., $ϵ=0.01$) or relaxed (e.g., $ϵ=0.1$). The concave behavior of the covert-rate curve reveals a fundamental trade-off between security enhancement and interference accumulation:

• Security Gain ($K<K^{★}$): In the low-K regime, increasing the number of active cooperative users enlarges the combinatorial selection space, which increases the structural variance ${\sigma }_X^2\left(K\right)$. This allows Alice to transmit with higher power while satisfying the covertness constraint, and the resulting signal gain dominates the additional interference at Bob.
• Interference Penalty ($K>K^{★}$): Beyond the optimal point, the aggregate interference at Bob grows faster than the security-driven transmit-power gain. Although the uncertainty at Willie continues to increase, the degradation in Bob’s received SINR dominates, leading to a reduction in the achievable covert rate.

Finally, Figure 5 compares the proposed public pilot-aided covert communication scheme with two representative benchmark methods in order to quantify the selection diversity gain. For a fair comparison, all schemes operate under the same covertness constraint, $ϵ=0.05$. In the best-jammer baseline, the total cooperative power $K{P}_c$ is concentrated on a single dominant user. The results reveal the following observations:

• Stability and scalability of the proposed scheme: The proposed method exhibits a smooth and monotonic increase in the achievable covert rate R as the number of cooperative users M increases. This behavior originates from the selection diversity gain. As the combinatorial search space $\left({\displaystyle \genfrac{}{}{0pt}{}{M}{K}}\right)$ grows, Bob can more reliably select a locally quiet subset that minimizes interference at the legitimate receiver while preserving high structural uncertainty at Willie.
• Volatility of single-user jamming: In contrast, the best-jammer baseline shows noticeable performance fluctuations across different values of M. This instability arises from the strong dependence on the spatial realization of a single jammer. Since the entire power budget is concentrated on one node, unfavorable geometric configurations (e.g., a jammer located close to Bob) can significantly degrade the received SINR, and no diversity mechanism exists to compensate for such cases.
• Inefficiency of random selection: The uniform random-selection baseline consistently achieves the lowest covert rate among the compared schemes. This result indicates that without intelligent subset selection, aggregate interference from multiple users primarily acts as performance degradation rather than a security resource. The comparison confirms that selection uncertainty, rather than mere interference accumulation, is the key factor enabling pilot-free covert communication.

7. Conclusions

In this paper, we investigated a public pilot-aided covert communication framework designed for next-generation wireless systems, with the objective of reducing reliance on pre-shared secret pilots. Unlike conventional covert communication schemes, the proposed approach exploits the selection uncertainty in multi-user cooperative networks to conceal the communication event. We analytically showed that even when Willie has full knowledge of individual channel state information (CSI), the structural uncertainty induced by the unknown active subset forces the optimal detector to reduce to an energy-based test. This result confirms that the uncertainty originates from the combinatorial structure of user selection rather than from channel estimation ambiguity. Theoretical analysis and numerical results demonstrated that the optimal transmit power $P_a^{*}$ scales with the square root of the selection-induced variance ${\sigma }_X^2$, which increases with the spatial dispersion in the network. In addition, the existence of an optimal active user size $K^{*}$ was identified, where the optimal value is governed primarily by network geometry and remains invariant with respect to the target covertness level $ϵ$. Benchmark comparisons further verified that the proposed selection-based cooperation achieves a significant selection diversity gain by simultaneously reducing interference at Bob and maintaining a robust masking effect against Willie. Overall, this work shows that spatial diversity and subset selection can serve as effective physical-layer resources for covert communication in public pilot-aided systems, eliminating the need for explicit secret pilot sharing.

It is worth noting that trusted execution environments (TEEs) [23] can partially address the secret pilot distribution problem by providing hardware-isolated enclaves for secure key management and remote attestation. However, TEE addresses the distribution of secrets and not the physical-layer overhead intrinsic to the secret pilot paradigm, such as pilot resource consumption and channel estimation latency. Moreover, TEE-based solutions require specialized hardware at every cooperative node and remain susceptible to side-channel vulnerabilities, which may limit their applicability in heterogeneous wireless networks. The proposed framework offers a complementary approach: while secret pilot-based schemes [15,16,17] achieve covertness through information asymmetry with well-established performance guarantees, the present work demonstrates that combinatorial subset uncertainty alone can also provide covertness, thereby broadening the design space for covert communication in public pilot-aided systems.

The proposed framework relies on three structural assumptions that, while appropriate for the centralized setting studied here, suggest concrete directions for future research. First, the current framework assumes centralized selection at a single receiver. Bob observes all M user-to-Bob channels via public pilots and deterministically selects a K-subset, producing a single layer of combinatorial ambiguity with $\left({\displaystyle \genfrac{}{}{0pt}{}{M}{K}}\right)$ candidates. In O-RAN architectures, the RAN Intelligent Controller (RIC) coordinates multiple distributed Radio Units (RUs), each serving a subset of users. This introduces a hierarchical selection structure: the near-RT RIC first partitions users across RUs, and each RU then performs local K-subset selection. Willie must resolve ambiguity at two levels—which partition and which local subset—resulting in a multiplicative increase in the combinatorial uncertainty space. A key research problem is to characterize the covertness gain from this hierarchical structure and to design joint partitioning-and-selection policies that maximize the aggregate covert throughput under fronthaul capacity constraints.

Second, in the current model, Willie observes a single aggregate interference signal, and the combinatorial uncertainty is therefore one-dimensional. In distributed MIMO systems, multiple geographically separated antenna panels jointly serve users. If Willie deploys multiple observation points or a multi-antenna receiver, the selection problem becomes multi-dimensional: each observation point sees a different weighted combination of the selected users’ signals due to distinct spatial signatures. A fundamental question is whether the additional spatial observations provide Willie with a resolvability gain that offsets the increased combinatorial uncertainty and how precoding design at the distributed panels can be optimized to maximize covert throughput.

Third, the current framework assumes that the network topology is fixed and fully known to Willie. In cell-free massive MIMO, a large number of access points (APs) are distributed over a wide area, and each user is served by a dynamically formed user-centric cluster of nearby APs. The active AP–user association changes across time slots based on large-scale fading conditions, which means the candidate set itself is no longer static. From Willie’s perspective, the uncertainty becomes two-fold: not only which K-subset of users is active but also which APs are serving them. A concrete research direction is to jointly optimize AP clustering and user selection to maximize covert throughput under the compound combinatorial uncertainty of AP–user association and subset selection.