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Covert Wireless Communication with Artificial Noise Generation

Ramin Soltani, Dennis Goeckel, Don Towsley, Boulat Bash, Saikat Guha

arXiv:1709.07096v5cs.ITcs.CRcs.NIeess.SP

TL;DR

Covert communication must hide that a transmission is occurring, not merely conceal its message content. This paper uses nearby friendly nodes to jam adversaries and shows that covert throughput scales with node density, while collaborating adversaries reduce achievable rates.

  • Problem

    Covert communication seeks to conceal transmission from an attentive adversary, beyond protecting only the message content.

  • Method

    The closest friendly node to each adversary generates artificial noise while other friendly nodes remain off.

  • Results

    Alice can reliably and covertly transmit O(min{n, m^γ/2√n}) bits in n channel uses, with no higher covert rate possible for γ > 2.

  • Takeaways & Limitations

    Randomly distributed friendly nodes can aid covert communication, whereas multiple collaborating adversaries inhibit it through increased interference.

Abstract

from arXiv · show

Covert communication conceals the transmission of the message from an attentive adversary. Recent work on the limits of covert communication in additive white Gaussian noise (AWGN) channels has demonstrated that a covert transmitter (Alice) can reliably transmit a maximum of $\mathcal{O}\left(\sqrt{n}\right)$ bits to a covert receiver (Bob) without being detected by an adversary (Warden Willie) in $n$ channel uses. This paper focuses on the scenario where other friendly nodes distributed according to a two-dimensional Poisson point process with density $m$ are present in the environment. We propose a strategy where the friendly node closest to the adversary, without close coordination with Alice, produces artificial noise. We show that this method allows Alice to reliably and covertly send $\mathcal{O}(\min\{{n,m^{γ/2}\sqrt{n}}\})$ bits to Bob in $n$ channel uses, where $γ$ is the path-loss exponent. Moreover, we also consider a setting where there are $N_{\mathrm{w}}$ collaborating adversaries uniformly and randomly located in the environment and show that in $n$ channel uses, Alice can reliably and covertly send $\mathcal{O}\left(\min\left\{n,\frac{m^{γ/2} \sqrt{n}}{N_{\mathrm{w}}^γ}\right\}\right)$ bits to Bob when $γ>2$, and $\mathcal{O}\left(\min\left\{n,\frac{m \sqrt{n}}{N_{\mathrm{w}}^{2}\log^2 {N_{\mathrm{w}}}}\right\}\right)$ when $γ= 2$. Conversely, we demonstrate that no higher covert throughput is possible for $γ>2$.

I. INTRODUCTION · II. SYSTEM MODEL, DEFINITIONS, AND METRICS · A. System Model

The paper studies covert wireless communication in which Alice hides message transmission from adversarial Willies, using friendly-node artificial noise in a Poisson wireless network. It establishes throughput limits for single and collaborating adversaries under AWGN channels, with the system model specifying node placement, transmission behavior, and path loss.

  • I. INTRODUCTION: Covert communication hides message existence, unlike secrecy methods that hide only message content, because detecting communication can penalize users.The motivation includes military operations, social unrest, and activity tracking.
  • I. INTRODUCTION: O(√n) bits is reliably transmissible from Alice to Bob in n AWGN channel uses while maintaining Willie’s detection-error constraint.Transmitting ω(√n) bits instead forces either Willie to detect Alice or Bob to incur a non-zero decoding-error probability asymptotically.
  • I. INTRODUCTION: The paper asks how covert throughput changes in wireless networks and introduces a single-hop scheme embedded in a large wireless network.The scheme considers both adversarial Willies that reduce throughput and friendly nodes that can increase it.
  • I. INTRODUCTION: O(min {n, m^(γ/2)√n}) bits are covertly transmissible in n channel uses by switching on the friendly node closest to Willie.The friendly nodes follow a two-dimensional Poisson point process with density m, and Alice and Bob share a codebook unknown to Willie.
  • I. INTRODUCTION: No algorithm for turning on friendly nodes enables Alice to transmit ω(m^(γ/2)√n) bits in n channel uses without detection or decoding reliability loss.A detector can either detect Alice with arbitrarily low error probability or prevent Bob from decoding with arbitrarily low error probability.
  • I. INTRODUCTION: With Nw collaborating Willies, closest-friendly-node artificial noise is applied near each adversary, but adversaries near Alice or Bob reduce covert throughput.A Willie near Alice forces lower transmission power, while a Willie near Bob experiences additional friendly-node noise that impairs decoding.
  • A. System Model: Alice communicates with Bob at unit distance amid Nw uniformly distributed Willies and friendly nodes modeled by a two-dimensional Poisson point process with density m.The model includes single-Willie and multiple-Willie scenarios, with all parties’ locations static and known.

B. Definitions · III. SINGLE WARDEN SCENARIO · A. Single Warden Scenario and γ > 2

The paper defines covertness through Willie’s detection error and reliability through Bob’s decoding error, then analyzes a single randomly located warden. Activating the friendly node closest to Willie enables O(min{n, m^γ/2√n}) covert bits in n channel uses, with a matching converse for γ > 2.

  • B. Definitions: For multiple collaborating Willies, the adversaries jointly process their received signals to make one collective decision about Alice’s transmission.The single-Willie case instead uses one hypothesis test on Willie’s received signal.
  • B. Definitions: Covertness requires lower-bounding Willie’s probability of error, while reliability requires Bob to decode with arbitrarily low error probability.Both definitions average over the random locations of friendly nodes and Willie(s).
  • III. SINGLE WARDEN SCENARIO: The single-warden scenario places one Willie uniformly and randomly on the unit square and distinguishes results for γ > 2 and γ = 2.Theorem 1.1 addresses γ > 2, while Theorem 1.2 addresses γ = 2.
  • III. SINGLE WARDEN SCENARIO: Activating the friendly node closest to Willie hides Alice’s transmission and enables O(min{n, m^γ/2√n}) covert bits in n channel uses.Alice and Bob turn on that closest node and keep the other friendly nodes off, regardless of whether Alice transmits.
  • A. Single Warden Scenario and γ > 2: The construction uses a secret shared codebook, Gaussian random coding, and maximum-likelihood decoding at Bob.Alice selects a new codebook for each message transmission, while the closest friendly node to Willie supplies artificial noise.
  • A. Single Warden Scenario and γ > 2: Choosing Alice’s average symbol power as O(m^γ/2√n) preserves covertness after averaging over friendly-node and Willie locations.Alice selects power and rate without using the realized node or Willie locations.
  • A. Single Warden Scenario and γ > 2: The converse holds regardless of which friendly nodes are active, and the closest friendly node’s generated noise dominates the noise from the other friendly nodes.Willie can use a threshold independent of friendly-node locations, and exceeding the converse power scale yields arbitrarily small average detection error.

B. Single Warden Scenario and γ = 2

With one uniformly located warden and γ = 2, friendly-node artificial noise enables Alice to covertly transmit O(min{n, m√n}) bits in n channel uses. If she attempts ω(m^(γ/2)√n) bits while only the closest friendly node is active, Willie can detect her or Bob may fail to decode reliably.

  • Single Warden Scenario and γ = 2: O(min{n, m√n}) bits are reliably and covertly transmissible in n channel uses when m > 0, γ = 2, and Willie is uniformly located over the unit square.This is the single-warden achievability result.
  • Single Warden Scenario and γ = 2: ω(m^(γ/2)√n) attempted bits exceed the covert limit when only Willie’s closest friendly node is active.The converse assumes Willie knows that only the closest friendly node is on.
  • Single Warden Scenario and γ = 2: At that attempted throughput, Willie can detect Alice with arbitrarily low error probability P(w) or Bob cannot decode the message with arbitrarily low error.The converse establishes that one of these failures must occur.
  • Single Warden Scenario and γ = 2: For γ = 2, the worst-case noise power is O(m log(m)), which is not optimal for the converse analysis.The proof therefore uses only the closest friendly node to Willie rather than all friendly nodes.

IV. MULTIPLE COLLABORATING WARDENS SCENARIO

With multiple collaborating Willies, Alice activates the closest friendly node to each Willie, enabling covert communication rates that depend on the path-loss exponent and number of wardens. For γ > 2, the achievable scaling is tight, while γ = 2 admits a distinct logarithmic penalty.

  • IV. Multiple Collaborating Wardens Scenario: Alice turns on the closest friendly node to each Willie and keeps all other friendly nodes off, regardless of whether she transmits.This strategy requires no close coordination between Alice and the friendly nodes.
  • Theorem 2.1: γ > 2: For γ > 2, Alice reliably and covertly sends O(min{n, m^(γ/2)√n/N_w^γ}) bits in n channel uses.Theorem 2.1 assumes m = ω(1), uniformly and independently distributed Willies, and N_w = o(m/log m).
  • Theorem 2.1: γ > 2: No higher covert throughput is possible for γ > 2 under the closest-friendly-node noise scheme.A detector based on the Willie closest to Alice establishes the converse.
  • Proof scope: When the number of collaborating Willies is finite, the analysis follows from the unbounded-N_w case, while the resulting O(1)-bit regime is excluded as uninteresting.The proofs explicitly assume N_w = ω(1).

V. DISCUSSION · A. Assumption of m = ω(1) in Theorems 2.1 and 2.2

Theorems 2.1 and 2.2 assume m = ω(1) to simplify the proof when Nw = ω(1), but this assumption can be relaxed with a corresponding condition on Nw. The growing-node assumption is also plausible for covert multi-hop communication in large wireless networks and aligns with related artificial-noise work.

  • A. Assumption of m = ω(1) in Theorems 2.1 and 2.2: m = ω(1) was assumed in Theorems 2.1 and 2.2 to simplify the proof when Nw = ω(1).
  • A. Assumption of m = ω(1) in Theorems 2.1 and 2.2: The assumption can be relaxed, but Nw = o(m/log m) must be replaced by Nw ≤ mζ 4 log (mζ/4).
  • A. Assumption of m = ω(1) in Theorems 2.1 and 2.2: m = ω(1) becomes plausible when the paper’s single-hop scheme is extended to covert multi-hop communication over large wireless networks.
  • A. Assumption of m = ω(1) in Theorems 2.1 and 2.2: In that extension, collections of nodes establish covert communication between multiple source and destination pairs.
  • A. Assumption of m = ω(1) in Theorems 2.1 and 2.2: The number of nodes often grows within a single communication hop as the overall network size grows.
  • A. Assumption of m = ω(1) in Theorems 2.1 and 2.2: The paper allows both friendly nodes and warden Willies to grow, with m = ω(1) and Nw = ω(1).
  • A. Assumption of m = ω(1) in Theorems 2.1 and 2.2: Related work analyzes key-less secure communication in a √n × √n cell using wireless fading dynamics and artificial noise from Poisson-distributed nodes of density one.

B. Assumption of turning on only the closest friendly node to each Willie

The achievability strategy activates only the friendly node closest to each Willie and keeps the others off; for a single Willie with γ > 2, the converse establishes this strategy as optimal. Although implementing it requires Willie locations, node collaboration, and deactivating many nodes, the paper argues these costs may be worthwhile to exceed O(√n) covert bits in n channel uses.

  • Assumption of turning on only the closest friendly node to each Willie: The proposed achievability strategy turns on only the friendly node closest to each Willie and keeps all other friendly nodes off.This assumption is used in the paper’s achievability proofs.
  • Assumption of turning on only the closest friendly node to each Willie: The converses of Theorems 1.2 and 2.1 are limited because they consider only strategies activating the friendly node closest to each Willie.The paper nevertheless considers this strategy likely optimal or close to optimal in practice.
  • Assumption of turning on only the closest friendly node to each Willie: Implementing the strategy requires knowing Willie locations, coordinating friendly nodes, and switching off many nodes, which may be costly.The paper argues that such costs can be reasonable in applications where covert communication is important, including military settings.
  • Assumption of turning on only the closest friendly node to each Willie: The strategy is motivated by increasing covert throughput beyond O(√n) bits in n channel uses.

C. High probability results · D. Assumption of uniform distribution for Willies · VI. CONCLUSION

The paper establishes covert communication rates aided by randomly distributed jamming nodes, examines uniform and Poisson models for Willie locations, and identifies converse results and remaining open problems. It also presents a high-probability covertness analysis for the single-Willie case.

  • C. High probability results: The covertness metric lower-bounds Willie’s expected probability of error over all instantiations of Willie and friendly-node locations.A high-probability covertness result for the single-Willie scenario is presented in Appendix M.
  • D. Assumption of uniform distribution for Willies: Conditioning a Poisson point process on the number of points in an area makes their locations uniformly distributed.This property motivates the uniform-location model used for the adversaries.
  • D. Assumption of uniform distribution for Willies: Theorems 1.1 and 1.2 model a single Willie uniformly on a unit box, while Theorems 2.1 and 2.2 use uniform locations for multiple Willies.The multiple-Willie model is chosen for consistency with the single-Willie scenario.
  • D. Assumption of uniform distribution for Willies: For γ > 2, modeling Willie locations by a Poisson process of rate λN produces results matching Theorem 2.1, except that λN is replaced by Nw.The Poisson-process analysis is given in Appendix N.
  • VI. CONCLUSION: O(min{n, m^γ/2√n}) bits are reliably and covertly transmitted to Bob in n channel uses when system nodes of density m aid in jamming Willie.The converse excludes higher covert rates for γ = 2 when the nearest node to Willie jams, and for γ > 2 without that assumption.
  • VI. CONCLUSION: Multiple collaborating adversaries inhibit communication by increasing effective SNR at their decision point and requiring more interference that impairs Bob’s decoding.These are identified as two separate effects in the conclusion.
  • VI. CONCLUSION: In the presence of Nw Willies, Alice can reliably and covertly send O bits when γ = 2.The supplied conclusion passage states this rate expression incompletely.
  • VI. CONCLUSION: Future work includes proving the converse for γ = 2 and embedding the single-hop results into large multi-hop covert networks.The conclusion also states that no higher covert throughput is possible for γ > 2 when the closest friendly node to each adversary transmits noise.

APPENDIX

The appendix proves the auxiliary inequalities and probabilistic bounds used in the main results. It also establishes conditions involving the path-loss exponent and derives bounds for collaborating adversaries.

  • A. Proof of (3): The proof of (3) compares ln(1 + x) with x − x^2 using a nonnegative derivative difference for x ≥0.The derivative difference is x^2, and 1 + x ≥0.
  • B. Proof of (7): The proof of (7) applies conditional expectations and the Poisson distribution of friendly nodes, using Willie’s location-independent noise characteristics.The result follows by combining the intermediate bounds and substituting c.
  • C. Proof of (14): The generalized Bernoulli inequality yields (1 + x)^−r ≤ (1 + rx)^−1 for x > −1 and r ≥1.The argument uses concavity of the logarithm and Jensen’s inequality.
  • E. Proofs of (23) and (24): The derivation of (23) requires γ > 2, while (24) follows by replacing γ with 2γ when γ > 2.The appendix explicitly notes that 2γ > 2 under this condition.

2. Thus,

The section proves intermediate probability and high-probability results using the WLLN, geometric bounds, and independence among node locations and Willie noise. It concludes that Alice’s average symbol power can be bounded by a term proportional to m^(γ/2).

  • 2. Thus,: The WLLN and independence of Willie locations support the limiting probability arguments used to establish the preceding result.The proof explicitly invokes the WLLN and independence between Willie and friendly-node locations.
  • 2. Thus,: Under N_w = o(m/log m), N_w = ω(1), and m = ω(1), the relevant probability limit is obtained.These growth conditions are stated before taking the limit.
  • 2. Thus,: The high-probability covertness proof uses independence between Willie noise and location together with conditional probability bounds.The proof assumes the locations of Willie and friendly nodes and separately uses σ_w^2 independence from d_a,w.
  • 2. Thus,: Alice can choose average symbol power P_a ≤ c m^(γ/2) in the high-probability argument.The passage states this power choice as part of the proof leading to the final bound.

N. Proof for the case where Willies are distributed according to a Poisson process: Instead

For Poisson-distributed friendly nodes and collaborating Willies with γ > 2, the proof activates the closest friendly node to each Willie while all others remain off. Under the stated density assumptions, this strategy supports reliable covert communication, and the converse follows by considering the closest Willie to Alice.

  • Construction: Alice and Bob activate the closest friendly node to each Willie and keep all other friendly nodes off regardless of transmission.The construction and Bob’s decoding are the same as in Theorem 2.1.
  • Assumptions: Theorem 2.3 assumes independently distributed friendly nodes and collaborating Willies with densities m = ω(1) and λN = o(m/log m), respectively.Alice and Bob are separated by unit distance.
  • Covertness: The covertness analysis bounds detection using Willies inside a finite-radius circle, then lets the radius tend to infinity.The proof uses the event that no Willie lies in a disk centered at Alice and extends the result through monotonicity.
  • Reliability: The reliability analysis shows that Bob’s decoding error probability can be made arbitrarily small under the stated Poisson density conditions.The argument accounts for artificial noise from nearby Willies and uses monotonicity in Bob’s noise power.
  • Number of Covert Bits: The proof establishes the corresponding number of covert bits Alice can send reliably in n channel uses.The result is identified in the theorem’s number-of-covert-bits analysis, although the supplied passage omits its displayed formula.
  • Converse: The converse uses the closest Willie to Alice as sufficient for detection and upper-bounds received noise by assuming all friendly nodes are active.The converse events are defined using the Willie density λN.

2. Thus,

The proof of (124) defines auxiliary events and bounds their probabilities using Poisson-process geometry and the weak law of large numbers. These bounds establish the required asymptotic conclusion.

  • Proof of (124): The proof of (124) begins by defining events and upper-bounding the probabilities of B′1, B′2, and B′3.These are identified as the next steps in the converse argument.
  • Proof of (124): For every Willie, the probability that the nearest friendly node is farther than δ is e−mπδ′2.This probability is identical across Willies and follows from (65).
  • Proof of (124): Under λN = o(m/log m), λN = ω(1), and m = ω(1), the derived probability bounds yield the stated limiting conclusion.The proof concludes after combining (145)–(147).
  • Proof of (124): The argument uses the expected number of Willies in a circle around Bob together with the weak law of large numbers.The cited passages specify a circle of radius λN and an average count of πλ2
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