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Covert Communication in the Presence of an Uninformed Jammer

Tamara V. Sobers, Boulat A. Bash, Saikat Guha, Don Towsley, Dennis Goeckel

arXiv:1608.00698v3cs.IT

TL;DR

The paper addresses whether positive-rate covert communication remains possible when Willie has a general receiver and an uninformed jammer assists Alice. It analyzes AWGN and block-fading channels, showing that Alice can use nonvanishing transmit power while remaining covert even against Willie’s optimal detector. Under AWGN links to Bob, this yields O(n) covert bits in n channel uses.

  • Problem

    Prior positive-rate covert communication results imposed receiver restrictions or did not establish achievability against Willie’s optimal detector, especially with multiple fading blocks.

  • Method

    The paper analyzes AWGN and block-fading channels with an uninformed jammer whose transmission is unknown to Willie during the detection period.

  • Results

    Alice can transmit with power not decreasing in n while remaining covert even when Willie employs an optimal receiver; for AWGN links to Bob, this implies positive-rate covert communication.

  • Takeaways & Limitations

    Uninformed jamming supports covert communication at positive rate under the stated AWGN conditions and under block-fading conditions against optimal detection.

  • Takeaways & Limitations

    The block-fading model assumes jammer power outside the codeword slot is independent of its power inside that slot, motivating future study of slot-boundary synchronism.

Abstract

from arXiv · show

Recent work has established that when transmitter Alice wishes to communicate reliably to recipient Bob without detection by warden Willie, with additive white Gaussian noise (AWGN) channels between all parties, communication is limited to $\mathcal{O}(\sqrt{n})$ bits in $n$ channel uses. However, this assumes Willie has an accurate statistical characterization of the channel. When Willie has uncertainty about such and his receiver is limited to a threshold test on the received power, Alice can transmit covertly with a power that does not decrease with $n$, thus conveying $\mathcal{O}(n)$ bits covertly and reliably in $n$ uses of an AWGN channel. Here, we consider covert communication of $\mathcal{O}(n)$ bits in $n$ channel uses while generalizing the environment and removing any restrictions on Willie's receiver. We assume an uninformed "jammer" is present to help Alice, and we consider AWGN and block fading channels. In some scenarios, Willie's optimal detector is a threshold test on the received power. When the channel between the jammer and Willie has multiple fading blocks per codeword, a threshold test on the received power is not optimal. However, we establish that Alice can remain covert with a transmit power that does not decrease with $n$ even when Willie employs an optimal detector.

I. INTRODUCTION

The paper studies covert communication aided by an uninformed jammer, extending positive-rate results beyond restricted Willie receivers to AWGN and block-fading settings. Its system model includes Alice, Bob, Willie, a continuously transmitting jammer, and slot-based observations.

  • Covert communication hides the existence of Alice and Bob’s exchange, not merely the message content, in settings including authoritarian surveillance and military operations.
  • The square root law limits reliable covert communication over AWGN channels to O(√n) bits in n uses when Alice and Bob share a sufficiently long secret.
  • Prior positive-rate results relied on Willie’s channel uncertainty and, in one AWGN setting, a receiver restricted to thresholding received power.
  • The paper introduces an uninformed jammer whose randomly varying Gaussian-noise power prevents channel estimation outside Alice’s transmission period from revealing Willie’s detection-period noise statistics.
  • For multiple fading blocks, total-power thresholding is sub-optimal, so the paper establishes covert transmission against Willie’s optimal detector rather than assuming a power detector.
  • The model gives Alice a possible n-symbol transmission slot, lets Willie use observations across T slots, and allows the uninformed jammer to transmit continuously under an average per-symbol power limit.

2) Block fading channels:

The paper models Rayleigh block fading with M independent fading blocks per codeword and analyzes Willie’s optimal hypothesis test using observations across slots. It establishes that M=1 permits an optimal power threshold test and constant-power covert transmission, whereas M>1 makes the total-power threshold sub-optimal.

  • 2) Block fading channels:: Rayleigh block fading remains constant for n/M symbols and changes independently across each of the M blocks per codeword.The fading coefficients are modeled for Alice- or jammer-to-Willie or Bob links, with different transmitter-receiver fading processes assumed independent.
  • B. Metrics, hypothesis testing, and likelihood ratio ordering: Willie tests H0 against H1 using observations from all slots, minimizing weighted false-alarm and missed-detection probabilities.H0 denotes no Alice transmission in slot t=0, while H1 denotes a message transmission there; the prior transmission probability p is known to Willie.
  • B. Metrics, hypothesis testing, and likelihood ratio ordering: With full statistical knowledge and simple hypotheses, Willie’s optimal detector is the likelihood ratio test.The paper uses likelihood-ratio ordering to derive monotonicity properties relevant to the detector structure.

III. AWGN CHANNELS

For AWGN channels, the paper combines random Gaussian codebooks with an uninformed jammer whose power varies independently across slots. This makes Willie’s optimal detector a total-power threshold test that becomes asymptotically ineffective, while Alice still achieves reliable covert transmission at positive rate.

  • III. AWGN CHANNELS: Alice and the jammer use a random-coding construction in which Alice’s Gaussian codebook is shared with Bob but unknown to Willie and the jammer.Alice transmits a selected length-n Gaussian codeword, while the jammer operates without knowing whether Alice transmits.
  • III. AWGN CHANNELS: Randomly varying jammer power across slots prevents Willie from using observations outside the target slot to estimate the interference statistics inside it.The jammer’s variance sequence is i.i.d. uniform on [0,Pmax], where Pmax is its maximum average power per symbol.
  • III. AWGN CHANNELS: Willie’s optimal detector is a threshold test on the total received power in the target slot.This is the detector structure established for the AWGN construction.
  • III. AWGN CHANNELS: The total received power in slot t=0 is a sufficient statistic for Willie’s AWGN test, and the likelihood ratio is non-decreasing in that statistic.The Fisher-Neyman factorization theorem yields sufficiency, while likelihood-ratio ordering establishes monotonicity.
  • III. AWGN CHANNELS: O(n) bits can be transmitted covertly and reliably in n AWGN channel uses with Alice’s transmit power not decreasing with n.The jammer’s interference at Bob is bounded so Bob’s received signal-to-noise ratio remains lower-bounded by a constant.
  • III. AWGN CHANNELS: For every threshold sequence chosen by Willie, Alice’s construction makes false alarms plus missed detections exceed 1−ε for sufficiently large n.Thus Willie’s detector becomes asymptotically useless under the paper’s covertness criterion.

A. Covertness with Transmit Power not Decreasing in the Blocklength

The paper analyzes how fading and jammer uncertainty shape Willie’s optimal detector. For AWGN and single-block fading, total received power thresholding is optimal, enabling Alice to transmit at power that does not decrease with blocklength.

  • Four channels are modeled: Alice-to-Bob, Alice-to-Willie, jammer-to-Bob, and jammer-to-Willie.
  • In the single-block fading case, Willie’s optimal detector compares total received power with a threshold.
  • When Willie lacks the Alice-to-Willie fading coefficient, granting him that knowledge still strengthens the achievability result.
  • A single-block fading strategy lets Alice transmit with power that does not decrease with blocklength while remaining covert.

B. The Number of Covert Bits Transmitted Reliably

The jammer changes the throughput implications of covert communication. Although block fading can prevent reliable O(n)-bit transmission under the strict reliability criterion, nonvanishing transmit power still improves outage-based performance.

  • O(n) bits are not strictly reliable over slowly fading Alice-to-Bob or jammer-to-Bob channels with M ≥ 1 blocks.For any constant R0 > 0, a nonzero probability of an unfavorable received SINR remains as n grows.
  • The jammer enables covert and reliable communication of O(n) bits in n uses when both Alice-to-Bob and jammer-to-Bob channels are AWGN.
  • Pf > 0 can remain independent of n when a jammer is present, versus O(1/n) power per symbol without one.
  • The analog of ε-outage capacity is non-zero in block-fading settings with the jammer.By contrast, it would be zero for Alice transmit power decreasing to 0 as n →∞.
  • For multiple fading blocks at Willie, total received power is not an optimal detector statistic by itself.The likelihood ratio instead uses the vector of received powers across blocks as a sufficient statistic.
  • With faded jammer-to-Willie channels, the likelihood ratio is monotonically increasing in each received-power component.

B. Covertness with Transmit Power not Decreasing in the Blocklength

For multiple fading blocks on the jammer-to-Willie link, the paper characterizes the optimal detector through received-power vectors rather than a scalar threshold. This structure supports covert transmission at power that does not decrease with blocklength.

  • For likely fading realizations, Willie’s detector is ineffective because no boundary curve works well across the relevant jammer-power values.Depending on the received jammer power, either missed detection or false alarm can be near one.
  • The optimal detector uses a boundary curve in the M-dimensional received-power space.The sufficient statistic is the vector of normalized average observed powers across blocks.
  • The jammer-to-Willie fading realization determines the expected jammer power separately in each block.
  • The proof selects δ and Pf so Alice’s received power remains below the detector’s boundary region with high probability.
  • Theorem 3 establishes a strategy in which Alice transmits with power that does not decrease with blocklength while remaining covert from Willie.

VI. DISCUSSION

The discussion situates jammer-assisted covert communication relative to square-root laws in covert communication and steganography, then identifies modeling assumptions that warrant further study.

  • Steganographic systems can obey square-root laws, although write-access to covertext can break that law without requiring uncertainty about Willie’s observations.The comparison distinguishes steganographic mechanisms from the jammer-assisted wireless setting considered here.
  • O(n) covert communication can be achieved with a jammer even when Willie uses an optimal receiver.The result holds for AWGN or block fading between the jammer and Willie, assuming an unlimited Alice–Bob key.
  • The block-fading assumption requires careful examination because jammer power outside the codeword slot is modeled as independent of power inside it.Randomly varying jammer power is suggested as a potential alternative if this model is too optimistic.
  • Small errors in synchronism between Alice’s and the jammer’s slot boundaries might allow Willie to estimate the environment and inhibit covert communication.The authors identify synchronism as an important assumption for future relaxation.

APPENDIX

The appendix analyzes covertness under AWGN and single-block fading, showing that constant Alice transmit power can satisfy the covertness constraint and identifying when Willie’s optimal detector is power-based.

  • For an AWGN Alice-to-Willie channel, Willie’s optimal receiver is a power detector when the jammer power is exponentially distributed.The detector compares the received-power statistic against a threshold.
  • For an M = 1 block-fading Alice-to-Willie channel, Willie’s optimal receiver is again a power detector when Willie knows the fading gain.The analysis assumes Willie knows h(a,w) for the relevant fading block.
  • The proof controls false alarms and missed detections by bounding jammer-power events and applying convergence and union-bound arguments for sufficiently large n.The resulting bounds combine at n greater than max(N0, N1).
  • The AWGN case concludes by applying arguments analogous to those used in Theorem 1.
  • Constant power Pf allows Alice to satisfy the covertness constraint for any ǫ > 0.The construction selects Pf after choosing the target average received power according to the AWGN analysis.

B. Proof of o(n) Covert Bits Transmitted for M = 1:

For the M = 1 fading model, the paper proves that Alice can transmit a sublinear number of covert bits while Bob decodes reliably under fading on all links.

  • o(n) bits can be transmitted covertly in n channel uses while Bob reliably decodes, when fading channels exist between all parties.
  • Alice uses constant power Pf > 0 independent of n while remaining covert, and the remaining task is to establish Bob’s decoding reliability.
  • Conditioned on h(a,b) and h(j,b), Bob’s channel is AWGN with a signal-to-noise ratio determined by the fading variables.
  • A constant rate R exists with sufficiently reliable communication, and o(n) is eventually smaller than nR, yielding the result.The argument uses the non-zero 2-outage capacity of the fading channel.

C. Proof of Increasing Λ(Z) for the M = 1 case for the Proof of Lemma 4:

The proof characterizes Willie’s observation through received power and shows that the likelihood ratio increases when the observed power increases beyond a boundary point.

  • Z = Σ_i=1^n |Zi|^2 is a sufficient statistic for Willie’s observation in the M = 1 fading model.The proof derives its distribution under both no transmission and Alice’s transmission.
  • Willie’s optimal decision rule is therefore expressed using the received-power statistic and the likelihood-ratio threshold.
  • The monotonicity argument compares likelihood-ratio integrals after extracting common integration terms and applying mean-value-theorem bounds.
  • If Λ(z(0)) = γ at a decision boundary, then increasing the observed power gives Λ(z) > γ.The proof establishes monotonicity of the likelihood ratio in the relevant observation.

D. Proof of Lemma 5

The proof constructs Rδ iteratively by expanding a codeword set through successive coordinate dimensions, adding δ-thickened boundary regions and bounding their probability with union bounds.

  • Construction: Rδ is built iteratively by starting with B1(n) and successively extending through dimensions 2,…,M.At each stage, the construction adds regions to the previously formed set.
  • Initialization: For fixed coordinates, the admissible first coordinate is either absent or unique, so the initialization uses no solution or a single solution for x1.The proof implicitly retains only points where g(x∼1) is defined.
  • Coordinate expansion: At stage m, fixing all coordinates except xm yields no solution, a single point, or an interval, and the construction adds thickness δ on both sides in dimension m.This extends the one-dimensional initialization to boundary regions across successive coordinates.
  • Coordinate expansion: The regions B1(n), B2(n), and BM(n) allow deviations of less than δ in one, two, and all M coordinates, respectively.Each later set preserves equality in coordinates not yet expanded.
  • Probability bound: A union bound combines the probabilities of the incrementally added regions, with the intermediate bound P(Um) ≤ δ supx fσ2j,1+σ2w(x) for m = 2,…,M.The proof repeats analogous bounding steps across dimensions and stages.
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