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Litter-Masked Block z-Channel: Designing Cover Distributions Against Public-Design Observers

Bin Han, Wenwen Chen, Yao Zhu, Rafael F. Schaefer, Giuseppe Caire, Hans D. Schotten

arXiv:2609.31767v1cs.ITeess.SPmath.IT

TL;DR

Silent idle slots let passive observers infer activity, while existing cover traffic leaves the upper layer with an erasure-only model. This paper shapes idle-slot litter from the codebook complement and couples that choice to a MAP receiver threshold. On a rate-3/4 (16, 12) LDPC instance, shaped litter reduces the observer’s Stein detection exponent by 41 to 48% relative to uniform cover, subject to a feasibility boundary below 10 dB.

  • Problem

    Silent idle slots can reveal operator activity, and the block z-channel validation based on a silent-idle null does not cover systems that replace that null with cover traffic.

    Source: §I · p. 1

  • Method

    The transmitter shapes a litter distribution on the codebook complement against a public-design observer, while a receiver threshold minimizes erasure under a silent-corruption cap.

    Source: §VI · p. 7, §VIII · p. 11

  • Results

    Shaped litter reduces the observer’s Stein detection exponent by 41 to 48% relative to uniform cover, with no operating point below 10 dB for the fixed-rate code.

    Source: §VIII · p. 12

  • Takeaways & Limitations

    Below 10 dB the fixed-rate code has no operating point, while a lower-rate or stronger code could trade spectral efficiency for a lower feasibility frontier.

    Source: D. Discussion and Limitations · p. 11, §VIII · p. 12

    At 8 and 10 dB, 20-event threshold calibration makes reported erasure uncertain by a factor of order 1.5, so the qualitative ordering is more reliable than the third digit.

    Source: D. Discussion and Limitations · p. 11

Abstract

from arXiv · show

Networking defenses against traffic analysis fill idle slots with cover traffic, yet the physical layer carrying it still hands upper layers the erasure-only model of a silent-idle link, leaving the cover content unexploited. We shape that content: the idle-slot waveform is drawn from the codebook complement under a design distribution, the \emph{litter distribution}, chosen to defeat a public-design passive observer while the upper layer keeps its block z-channel model. The maximum a posteriori (MAP) receiver reduces to a single threshold on the log-likelihood ratio between the best codeword and the prior-weighted litter aggregate; its errors are silent corruptions the upper layer cannot detect and erasures that retransmission absorbs. Receiver design is thus one-dimensional, minimizing erasure under a cap on silent corruption, and the joint design a Stackelberg game against a Kullback-Leibler (KL) indistinguishability metric, solved by an alternating convex relaxation and corroborated by an independent reinforcement-learning policy. On a rate-$3/4$ $(16,12)$ parity-check (LDPC) code against an observer who knows her own channel but whose channel the transmitter knows only statistically, shaped litter cuts the observer's Stein detection exponent by $41$ to $48\%$ relative to uniform cover, flat over a $10$~dB span and in the activity rate. Below the point where the legitimate link turns reliable, no cover distribution admits an operating point; at that point shaping costs about one percent of throughput, and nothing measurable $2$~dB above it.

I. INTRODUCTION

The paper replaces silent idle slots with cover drawn from the codebook complement, shaping its distribution against a public-design observer while retaining the upper layer’s block z-channel model. It frames receiver-threshold selection and transmitter-side litter shaping as coupled design problems and evaluates the approach against related networking and physical-layer defenses.

I. INTRODUCTION

  • The litter-masked design draws idle-slot sequences from the codebook complement and jointly shapes their distribution and the receiver threshold to limit observer distinguishability and link errors.The threshold minimizes erasure under a cap on silent corruption, while litter shaping minimizes the public-design observer’s distinguishing power subject to the receiver’s erasure budget.Source: §I · pp. 1–2

II. RELATED WORK

  • Prior approaches shape cover at packet or symbol level, assume silent idle slots, use the codebook itself as covert payload, or embed cover in active waveforms.This paper instead shapes a per-block distribution on the codebook complement and couples it to the decoder threshold.Source: §I · p. 2, §II · p. 2

I. INTRODUCTION

  • On a rate-3/4 (16, 12) LDPC instance over 165 independent runs, shaped cover removes 41 to 48% of the observer’s detection exponent wherever the legitimate link has an operating point.The study evaluates the design against an observer whose channel is known to the observer but modeled statistically by the transmitter.Source: §I · p. 2

III. SYSTEM MODEL

  • The system uses a block codebook whose complement is the pool of non-codeword sequences available for litter.Both the codebook and litter set are public knowledge.Source: §III · p. 2

A. Transmitter

  • Activity is Bernoulli with probability p; active blocks use a uniform codeword, while idle blocks follow the designable litter distribution.Uniform litter is the no-shaping baseline.Source: A. Transmitter · p. 3

B. Channel and Observation

The model sends either a codeword or litter in every slot over a coherent AWGN channel, then groups receiver outcomes by what the upper layer observes. It treats undetected packet-like errors as silent corruption and detected losses as erasures, with distinct constraints for each.

B. Channel and Observation

  • The numerical study uses Gray-labeled QPSK over a block-constant complex-gain channel with AWGN, known gain at the receiver, and no transmitter–receiver shared secret.The analysis applies to any labeling, while the coset-class construction uses Gray-QPSK geometry.Source: B. Channel and Observation · p. 3

C. Operational Event Taxonomy

  • The six elementary transitions reduce to five link-level outcomes because relabeling one litter sequence as another is indistinguishable from correct idleness.The outcomes are correct decoding, correct idleness, codeword confusion, erasure, and false alarm.Source: C. Operational Event Taxonomy · p. 3

D. Operational Error Metrics

  • Codeword confusion and false alarms are silent corruptions, whereas erasures are detected losses recoverable by retransmission or outer erasure coding.The receiver minimizes erasure subject to a silent-corruption cap and an erasure budget; jointly optimizing outer-code rate across all three error events is left open.Source: D. Operational Error Metrics · pp. 3–4

IV. OPTIMAL MAP DECODER

The MAP receiver compares the best codeword likelihood with the prior-weighted aggregate likelihood of litter, yielding a single log-likelihood-ratio threshold. The threshold remains a design variable so the receiver can trade false alarms against erasures under operational caps.

IV. OPTIMAL MAP DECODER

  • The MAP decoder reduces to a threshold on the log-likelihood ratio between the best-codeword likelihood and the prior-weighted litter aggregate.The receiver knows the codebook, litter set, activity prior, and litter distribution.Source: §IV · p. 4, A. Likelihood Statistics · p. 4

A. Likelihood Statistics

  • The likelihood statistic compares a maximum over codewords with an aggregate over litter, matching individual codeword decisions to class-level idleness decisions.The codeword-wise inner decoder selects the codeword closest to the observation in Euclidean distance.Source: A. Likelihood Statistics · p. 4

B. Bayes-Optimal Decision Rule

  • Under merged-outcome zero-one loss, the Bayes-optimal receiver chooses a codeword when its posterior risk is below the risk of declaring litter.The resulting threshold is biased toward idleness in proportion to the prior mass assigned to litter.Source: B. Bayes-Optimal Decision Rule · p. 4
  • Full-sequence maximum likelihood compares the best codeword with only the best litter and ignores the activity and litter priors, so it is not generally Bayes-optimal.Codeword-only ML never declares idleness and therefore labels every litter slot a false alarm.Source: B. Bayes-Optimal Decision Rule · p. 4

C. Neyman-Pearson Interpretation and Receiver Design Parameter

At fixed litter distribution, receiver design minimizes erasure under a hard cap on silent corruption. Because the error rates move monotonically with the threshold, the cap selects the operating point on the receiver’s trade-off curve.

C. Neyman-Pearson Interpretation and Receiver Design Parameter

  • Receiver design follows the Neyman-Pearson direction: minimize erasure while capping silent corruption, whose false-alarm component is the Type-I error.Idle is H0 and active is H1; erasure is the Type-II error.Source: C. Neyman-Pearson Interpretation and… · p. 5
  • As the threshold rises, false alarms and silent corruption decrease while erasures increase; codeword confusion is bounded by the codeword-ML pair-confusion probability.That pair-confusion contribution depends on codebook geometry and SNR, not on the threshold.Source: C. Neyman-Pearson Interpretation and… · p. 5
  • The optimal threshold is the smallest value meeting the silent-corruption cap, and its dependence on the litter distribution couples receiver design to transmitter shaping.Sweeping the threshold traces the false-alarm–erasure trade-off at fixed codebook, SNR, and litter distribution.Source: C. Neyman-Pearson Interpretation and… · p. 5

V. THREAT MODEL AND INDISTINGUISHABILITY METRIC

The threat model gives Eve public knowledge of the design and a passive, multi-block view, while the transmitter knows Eve’s channel only statistically. The analysis accounts for slow channel variation that can sometimes reverse Bob’s median link advantage.

A. Observer Model

  • Bob reports his instantaneous channel state to the transmitter, but the transmitter knows Eve’s channel only through an SNR prior.Eve estimates her own gain from the shared pilot preamble; neither party observes the other’s gain block by block.Source: A. Observer Model · p. 5
  • Independent shadowing overturns a 6 dB median advantage for Bob in about one realization in six when σ = 6 dB.The design therefore accounts for realizations in which Eve’s link is better placed.Source: A. Observer Model · pp. 5–6
  • Eve is modeled as a passive observer who knows the codebook, activity prior, and litter distribution and may observe multiple blocks.She does not jam or otherwise interfere with the channel.Source: A. Observer Model · p. 6

B. Eve’s Hypothesis Tests

Eve’s activity inference is framed as a per-block likelihood-ratio test, while traffic analysis tests whether the full observation window is all idle or drawn from the activity mixture.

B. Eve’s Hypothesis Tests

  • Eve’s observations are conditioned on activity and her SNR, with active and idle output mixtures defining the alternatives.The per-block likelihood ratio is formed from those alternatives.Source: B. Eve’s Hypothesis Tests · p. 6
  • For a given block, Eve’s posterior activity decision depends only on that block’s observation, so processing the full sequence does not improve the likelihood-ratio test.This follows from i.i.d. activity states and conditional independence of blocks given Eve’s SNR.Source: B. Eve’s Hypothesis Tests · p. 6
  • The window-level traffic-analysis test compares the all-idle product distribution with the activity-mixture product distribution at Eve’s realized SNR.Source: B. Eve’s Hypothesis Tests · p. 6

C. Indistinguishability Metric

The paper uses expected KL divergence to measure how distinguishable idle and active outputs are to Eve, then implements the SNR-dependent design through a CSI-indexed lookup table.

C. Indistinguishability Metric

  • The expected per-block KL divergence is the design metric because it bounds Eve’s presence-test exponent across activity rates.Stein’s lemma relates that exponent to the number of blocks needed to attain a target miss probability.Source: C. Indistinguishability Metric · p. 6
  • The metric is an expected exponent, not the exponent of Eve’s expected error; long-window average miss probability is dominated by the weakest SNR realization.A guarantee against every realization requires bounded support for Eve’s SNR prior.Source: C. Indistinguishability Metric · p. 6

D. CSI-Feedback Adaptation Architecture

  • The CSI-feedback implementation stores litter distributions and receiver thresholds by Bob’s SNR, then indexes the table at block boundaries without online optimization.The numerical study holds the coding scheme fixed and varies SNR.Source: D. CSI-Feedback Adaptation Architecture · p. 7

VI. OPTIMAL LITTER DISTRIBUTION SHAPING

The transmitter selects a litter distribution to reduce Eve’s distinguishing power, anticipating the receiver’s threshold best response under the error caps. The resulting joint problem is non-convex and is approached with two solvers that lack a global-optimum guarantee.

VI. OPTIMAL LITTER DISTRIBUTION SHAPING

  • The Stackelberg design has the transmitter minimize indistinguishability while the receiver best-responds with the erasure-minimizing threshold under the silent-corruption cap.The design also requires the resulting erasure rate to remain within the retransmission budget.Source: §VI · p. 7, A. Stackelberg Formulation · p. 7

A. Stackelberg Formulation

  • Interchanging which error cap is enforced by the receiver and transmitter leaves the feasible litter distributions and optimizer unchanged, though the operating threshold differs.The paper solves the erasure-capped ordering and reports both operating points.Source: A. Stackelberg Formulation · p. 7
  • The joint program is non-convex because the MAP statistic and receiver threshold depend on the litter distribution; the convex-relaxation and DRL solvers have no global-optimum guarantee.The paper uses their agreement as evidence that neither solver is far from the achievable optimum.Source: A. Stackelberg Formulation · p. 7

B. Decomposition and Alternating Convex Relaxation

The alternating relaxation freezes the receiver response to obtain a convex litter-shaping subproblem, then updates the receiver threshold and resolves the subproblem iteratively. The resulting feasible design provides an upper bound on the joint problem’s optimum, but the method makes no stationarity or global-optimum guarantee.

B. Decomposition and Alternating Convex Relaxation

  • The receiver’s best response is recomputed at every outer iterate because the likelihood-ratio test statistic changes with the litter distribution.Source: B. Decomposition and Alternating… · p. 7
  • The frozen-reference shaping step is a convex program on the litter-distribution simplex with one affine constraint, using the reference threshold to enforce the erasure cap.Convexity follows from freezing the test statistic and receiver rule at the reference distribution.Source: B. Decomposition and Alternating… · pp. 7–8
  • At each outer iteration, the algorithm updates the reference, recomputes the receiver threshold, and re-solves the shaping problem within a chi-square trust region.In experiments, the trust-region distance is capped at 1, guaranteeing an effective sample fraction of at least one half.Source: B. Decomposition and Alternating… · p. 8
  • The alternating iterates decreased the objective monotonically in the reported experiments, but the method claims no stationarity or global-optimum guarantee.The converged distribution is feasible by construction and supplies a documented upper bound on the optimum.Source: B. Decomposition and Alternating… · p. 8
  • A defensive mixture with uniform mass α = 0.05 prevents importance-sampling divisors from exploding as the reference distribution becomes sparse.Without the mixture, the largest divisor reached 8.5 × 10^6 by iteration five and the solver failed; with it, every iteration completed.Source: B. Decomposition and Alternating… · p. 8

C. Deep-Reinforcement-Learning Solver

The paper also solves the joint design directly with a reinforcement-learning policy, providing an independent check on the alternating relaxation. PPO selects litter distributions while the receiver threshold is computed by binary search, and the learned policy converges only to a local stationary point.

C. Deep-Reinforcement-Learning Solver

  • The DRL policy solves the joint problem directly, without freezing the test statistic or imposing a convex-program reformulation, making it an independent check on the alternating relaxation.The two methods share no approximation, so agreement between them is informative.Source: C. Deep-Reinforcement-Learning Solver · p. 8
  • PPO [46] is used for the one-step contextual bandit because it supports stochastic simplex-valued actions at moderate action dimensions.The solver uses a Dirichlet policy head in place of PPO’s biased Gaussian-softmax head on the simplex.Source: C. Deep-Reinforcement-Learning Solver · p. 8
  • The action is a class-restricted litter distribution, and each reward evaluation computes the receiver’s threshold by binary search at the erasure cap.The reward penalizes silent corruption at that threshold; the threshold itself is computed, not learned.Source: C. Deep-Reinforcement-Learning Solver · p. 8
  • The policy restricts outputs to the class simplex, reducing the action dimension from the full litter set to a moderate number of classes.The policy warm-starts from the alternating iterate and refines a feasible design rather than re-exploring from uniform.Source: C. Deep-Reinforcement-Learning Solver · p. 9
  • The learned policy converges to a local stationary point, not a guaranteed global optimum of the joint design problem.Source: C. Deep-Reinforcement-Learning Solver · p. 9

D. Practical Computation

Practical computation depends on restricting litter distributions to equivalence classes and approximating the observer expectation with sampled channel and output realizations. For the high-rate (16, 12) geometry, coset classes make the restriction tractable and permit efficient likelihood evaluation.

D. Practical Computation

  • Class restriction reduces optimization from the full litter simplex, whose dimension is M^n − M^k, to a C-dimensional simplex with a distribution uniform within each class.The partition is rate-adaptive, and the class-level false-alarm coefficients aggregate the litter-level coefficients.Source: D. Practical Computation · p. 9
  • For the (16, 12) code, 255 coset classes are geometry-homogeneous, and each alternating iteration solves in seconds.Class membership requires only a syndrome computation, and the classes avoid enumerating the codebook.Source: D. Practical Computation · p. 9
  • The observer expectation and output integral are estimated using K channel draws and empirical output samples, with bias controlled by the draw and sample counts.In the high-rate regime, the active-side mixture likelihood is evaluated exactly in O(M^(n−k)) per observation using Poisson summation over the dual code.Source: D. Practical Computation · p. 9
  • Without class restriction, both the alternating algorithm and DRL solver are intractable at litter-set sizes near 10^9 on practical codes.Source: D. Practical Computation · p. 9

VII. NUMERICAL RESULTS

On a rate-3/4 (16, 12) LDPC instance, shaped litter reduces the observer’s detection exponent by 41–48% above 10 dB, with the reduction flat across SNR and activity rate. Below 10 dB no cover distribution provides an operating point; at 10 dB shaping costs about one percent of throughput, and from 12 dB upward the reduction has no measurable throughput cost.

A. Experimental Setup

  • The study uses a rate-3/4 (16, 12) LDPC code, sweeps Bob’s SNR and activity rate, and models Eve’s SNR with a log-normal prior.The fixed code isolates cover-design sensitivity to link quality; it is not a deployment profile because real links adapt coding and modulation to SNR.Source: A. Experimental Setup · pp. 9–10, §VII · p. 9

B. Indistinguishability Reduction

  • Below 10 dB shaping buys nothing: the design is infeasible up to 4 dB, while at 6 and 8 dB heavy mixture overlap leaves ¯D near uniform.No cover distribution admits an operating point below 10 dB.Source: B. Indistinguishability Reduction · p. 10, Figure 1 · p. 10
  • From 10 dB upward, shaped cover reduces ¯D by 41–48% versus uniform cover, consistently across SNR and activity rate.The reduction remains flat over a 10 dB span even as ¯D grows by a factor of 14.Source: B. Indistinguishability Reduction · p. 10, Figure 1 · p. 10, Table II · p. 10
  • Across 18 cells from 10 dB upward, DRL trails the alternating relaxation by 0.3–1.4 percentage points, always within one standard error.The two solvers share no approximation, so their agreement corroborates the result; DRL serves as corroboration rather than improvement.Source: B. Indistinguishability Reduction · p. 10
  • One 20 dB, p = 0.1 cell measured a 38% reduction with a standard error four times its neighbors’, while all runs from 6 dB upward completed.The 20 dB cell is reported as measured; the other four seeds in that cell yielded 43–52%.Source: B. Indistinguishability Reduction · p. 10, Table II · p. 10

C. Operational Compliance

  • From 0 to 6 dB no threshold or cover distribution meets both caps; at 8 dB shaped cover cuts erasure from 11% to 1.7–1.9%, but remains infeasible.Codeword confusion exceeds the silent-corruption cap at 0–6 dB; at 8 dB the cover begins to matter despite the lack of a feasible operating point.Source: C. Operational Compliance · p. 11
  • At 10 dB shaping cuts ¯D by 41% and costs about one percent of throughput; shaped designs’ erasure is marginally above the cap and within calibration noise.Uniform cover is feasible at 1.3 × 10^-3 erasure, while shaped designs sit at 1.3–1.4 × 10^-2.Source: C. Operational Compliance · p. 11
  • From 12 dB upward, deployed erasure is below 2 × 10^-5 for every method and unmeasurable from 14 dB, so shaping’s ¯D reduction costs no measurable throughput.The feasibility frontier below 10 dB is set by codeword confusion, not the cover distribution.Source: C. Operational Compliance · p. 11

D. Discussion and Limitations

The conclusion reports that shaped litter reduces the observer’s detection exponent while the fixed-rate code has no operating point below 10 dB. The discussion qualifies this result by the observer-channel model, calibration accuracy, and implementation limits, and identifies broader settings for future work.

D. Discussion and Limitations

  • The feasibility frontier is specific to this code and these SNRs; lower-rate or stronger codes could lower it, while implementation optimizations bring an operating point under half an hour.Source: D. Discussion and Limitations · p. 11
  • At 8 and 10 dB, 20-event threshold calibration makes reported erasure uncertain by a factor of about 1.5, although the qualitative ordering is robust.Source: D. Discussion and Limitations · p. 11
  • At high Bob SNR, the expected exponent is dominated by high-Eve-SNR realizations where no litter distribution protects; worst-case reporting would require bounding Eve’s SNR.Source: D. Discussion and Limitations · p. 11
  • Below 6 dB the relaxation is infeasible because codeword confusion exceeds the silent-corruption cap; deployment should fall back to uniform rather than the learned policy.Source: D. Discussion and Limitations · p. 11

VIII. CONCLUSION

  • The framework uses a single-threshold MAP receiver and Stackelberg litter shaping, solved by alternating convex relaxation and independently by a PPO policy.Source: §VIII · p. 11
  • 41–48% lower Stein detection exponent than uniform cover, flat above 10 dB and across activity rates, with both solvers agreeing within one standard error.Source: §VIII · p. 12
  • Below 10 dB this fixed-rate code has no operating point; shaping costs about one percent of blocks at 10 dB and no measurable erasure from 12 dB upward.Source: §VIII · p. 12
  • Future work includes standards-length codes, observer-position constraints for worst-case design, finite-blocklength bounds, outer-code objectives, and richer threat models.Source: §VIII · p. 12

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