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Legitimate Reconstruction Advantage in Wiretapped Semantics-Aware Remote Monitoring

Bowen Li, Nikolaos Pappas

arXiv:2610.21819v1cs.ITmath.IT

TL;DR

Remote monitoring over wireless links must support accurate reconstruction for an intended receiver while limiting what an eavesdropper can reconstruct. This paper defines a joint LRA measure and analyzes it under randomized stationary transmission and zero-order-hold reconstruction. Its results show that policies favoring LRA probability can differ from those favoring persistence, and that improving legitimate accuracy need not improve LRA probability.

  • Problem

    Because source dynamics affect whether received updates remain correct, marginal reconstruction accuracies do not generally determine the joint event of legitimate success and eavesdropper failure.

    Source: §I · p. 1

  • Method

    The paper defines LRA as that joint event and uses joint stationary analysis to derive its probability and persistence under randomized stationary transmission and zero-order-hold reconstruction.

    Source: §I · p. 2, 2) Transmission Policy · p. 2

  • Results

    LRA probability and persistence can favor different policies, marginal-objective optimization can be suboptimal for LRA, and improved legitimate-channel reliability need not increase LRA probability.

    Source: §I · p. 2, B. Rare-Transmission Limits · p. 5, 3) Effect of Legitimate-Channel Quality · p. 8

  • Takeaways & Limitations

    Transmission decisions should account for both receivers’ reconstruction outcomes and the monitoring application’s temporal requirements.

    Source: §VII · p. 9

    The conclusions are scoped to randomized stationary transmission and prescribed zero-order-hold reconstruction.

    Source: §VII · p. 9

Abstract

from arXiv · show

This paper investigates joint accuracy and security in semantics-aware remote monitoring, where source updates intended for a legitimate receiver may also be intercepted by an eavesdropper. We define legitimate reconstruction advantage (LRA) as the joint event that the intended receiver correctly reconstructs the current source state using received updates while the eavesdropper reconstructs it incorrectly. Source evolution changes the relevance of received updates at the two receivers, so their marginal reconstruction accuracies do not, in general, determine the probability of this joint event. Under randomized stationary (RS) transmission and zero-order-hold reconstruction at both receivers, we derive an explicit joint stationary distribution and closed-form expressions for LRA probability and persistence for general finite-state ergodic Markov sources. For two-state and uniform-transition sources, we obtain algebraic expressions that enable low-complexity optimization of RS policies. Analytical and numerical results show that maximizing the probability of legitimate advantage need not maximize its persistence. For two-state sources, rare-transmission analysis reveals that, at a fixed sum of transition probabilities, greater transition asymmetry can increase persistence while reducing LRA probability. Moreover, optimizing marginal accuracy and security metrics can select policies that are suboptimal for LRA. Counterintuitively, more frequent transmission or improved legitimate-channel reliability can increase legitimate reconstruction accuracy while reducing LRA probability. Finally, a geofencing application translates prescribed LRA requirements into spatial constraints on eavesdropper locations.

I. INTRODUCTION

The paper frames remote monitoring under eavesdropping as a joint accuracy-and-security problem because source dynamics make receiver-wise metrics insufficient to determine simultaneous reconstruction outcomes. It introduces LRA analysis under randomized stationary transmission and zero-order-hold reconstruction, then describes the system assumptions and model.

I. INTRODUCTION

  • Source dynamics can make marginal receiver metrics insufficient to determine whether Bob reconstructs correctly while Eve reconstructs incorrectly.A staler observation may be correct while a fresher one is not, so age-based asymmetry alone does not determine joint accuracy and security.Source: §I · p. 1
  • The paper defines LRA as Bob’s reconstruction success jointly with Eve’s failure and derives closed-form probability and persistence metrics for finite-state ergodic Markov sources.For two-state and uniform-transition sources, algebraic expressions support rare-transmission analysis and policy optimization; numerical results also show that LRA probability and persistence can favor different policies, while geofencing translates LRA requirements into spatial constraints.Source: §I · p. 2

II. SYSTEM MODEL

  • The modeled wiretap system has Alice transmitting source updates to Bob and Eve, with the objective of accurate reconstruction at Bob and incorrect reconstruction at Eve.Fig. 1 depicts the transmitter, legitimate receiver, and eavesdropper.Source: §II · p. 2, Figure 1 · p. 2

1) Source Process:

2) Transmission Policy:

  • Under an RS policy, Alice’s transmission decisions are i.i.d. Bernoulli variables independent of the source process.The transmission probability is denoted by p_a.Source: 2) Transmission Policy · p. 2

3) Erasure Wiretap Channel:

  • The links to Bob and Eve are modeled as erasure channels, with independent channel-state processes that are i.i.d. over time and independent of the source and transmission decisions.A receiver obtains the current source state only when Alice transmits and its link successfully delivers the update.Source: 3) Erasure Wiretap Channel · p. 2, 4) Receiver Observation and… · p. 3

4) Receiver Observation and Reconstruction:

  • Both receivers use zero-order hold, retaining their latest successfully received source state until another update arrives.Before a receiver’s first successful reception, it retains an arbitrary initial reconstruction.Source: 4) Receiver Observation and… · p. 3

5) Adversary Model:

The analysis treats Eve as a passive interceptor and defines LRA through the receivers’ simultaneous reconstruction outcomes. A joint Markov-state model then supports stationary analysis of the LRA metrics.

5) Adversary Model:

  • Eve is passive and uses the same zero-order-hold reconstruction rule as Bob; security is defined as Eve’s reconstruction failure under that rule, without assuming it is optimal for Eve.Eve does not modify the source, transmission decisions, or channel conditions.Source: 5) Adversary Model · p. 3

B. Legitimate Reconstruction Advantage Metrics

  • LRA probability measures how often Bob is correct while Eve is incorrect, whereas LRA persistence measures the average duration of the current consecutive advantage run.The two metrics capture distinct aspects of joint performance, so optimizing one does not generally optimize the other.Source: B. Legitimate Reconstruction Advantage… · p. 3

III. GENERAL PERFORMANCE CHARACTERIZATION

  • The general characterization derives a stationary distribution for the joint state of the source and both reconstructions, then uses it to obtain closed-form LRA probability and persistence.The joint state tracks the source together with Bob’s and Eve’s reconstructions.Source: §III · p. 3, A. System Analysis · pp. 3–4

A. System Analysis

  • The coupled source and receiver reconstructions form a finite-state, time-homogeneous Markov chain with a unique aperiodic recurrent class and stationary distribution.Its long-term behavior is independent of the initial condition, enabling stationary expectations for the LRA metrics.Source: A. System Analysis · p. 3

B. Metric Analysis

The paper expresses long-term LRA probability and persistence through stationary quantities, then specializes the general analysis to structured source models. Two-state and uniform-transition cases yield algebraic forms for further analysis.

B. Metric Analysis

  • The general LRA probability and persistence are computed from the restricted stationary vector and transition submatrix, and both metrics are rational functions of transmission and channel-success probabilities.These computable metrics provide a basis for parameter optimization.Source: B. Metric Analysis · p. 4

IV. STRUCTURED-SOURCE SPECIALIZATIONS

  • For two-state and uniform-transition Markov sources, the general matrix-form results specialize to explicit algebraic expressions for LRA probability and persistence.These forms clarify effects of source dynamics and transmission decisions and support rare-transmission analysis.Source: §IV · p. 4

1) Two-State Markov Source:

  • For the two-state source, separate corollaries give algebraic expressions for long-term average LRA probability and persistence.The supplied passages identify the expressions but do not state their formula values.Source: 1) Two-State Markov Source · p. 4

2) Uniform-Transition Markov Source:

For a uniform-transition source, state symmetry gives a uniform stationary distribution, and the paper derives algebraic LRA metrics. In the rare-transmission limit, LRA probability and persistence share a receiver-separation factor but depend on different aspects of source dynamics.

2) Uniform-Transition Markov Source:

  • The uniform-transition source stays in its current state with probability ρ and moves to each other state with probability η, yielding a uniform stationary distribution.The transition probabilities satisfy ρ + (N − 1)η = 1.Source: 2) Uniform-Transition Markov Source · p. 5
  • Corollaries 4 and 5 give algebraic expressions for the uniform-transition source’s long-term average LRA probability and persistence.The expressions specialize the general results to a symmetric state-transition structure.Source: 2) Uniform-Transition Markov Source · p. 5

B. Rare-Transmission Limits

  • In the rare-transmission limit, LRA probability and persistence share a channel-induced receiver-separation factor, while their source-dependent factors capture state mismatch and mismatch-weighted sojourns, respectively.Numerical experiments suggest this factorization also holds for general Markov sources.Source: B. Rare-Transmission Limits · p. 5
  • For two-state sources at fixed transition sum, balanced transitions maximize the rare-limit LRA probability factor, whereas greater transition asymmetry increases the persistence factor.The probability-side factor depends on state mismatch; the persistence-side factor depends on mismatch-weighted sojourns.Source: B. Rare-Transmission Limits · pp. 5–6

V. TRANSMISSION PROBABILITY OPTIMIZATION

The paper optimizes RS transmission probability for either LRA probability or persistence using general-source expressions and simpler structured-source formulas. The optimization reduces to evaluating boundary points and stationary candidates, with a closed-form policy characterization in the symmetric-channel two-state case.

V. TRANSMISSION PROBABILITY OPTIMIZATION

  • The paper optimizes RS transmission probability for LRA probability or persistence, using numerical optimization for general sources and polynomial root finding for structured sources.For two-state LRA probability, the paper also provides closed-form solutions.Source: §V · p. 6, A. Optimality Analysis · p. 6

A. Optimality Analysis

  • The optimization extends the objective continuously to pa = 0 using its rare-transmission limit, which is not the performance of a zero-transmission policy.If the extended optimum occurs at zero, the original supremum is approached as pa tends to zero from above.Source: A. Optimality Analysis · p. 6
  • For a rational objective, Theorem 4 identifies the maximizer by comparing the endpoints with interior roots of a scalar polynomial equation.Interior optima satisfy the derivative condition PF(pa) = 0.Source: A. Optimality Analysis · p. 6
  • With a symmetric channel, the optimal transmission probability depends entirely on source temporal correlation: persistent sources favor sparse transmission, alternating sources full transmission, and i.i.d. sources are indifferent.The objective is increasing when p + q > 1, decreasing when p + q < 1, and constant when p + q = 1.Source: A. Optimality Analysis · p. 6

B. Complexity Analysis

The analysis reduces optimization complexity for general and structured sources, then validates its metrics and shows that probability, persistence, and marginal objectives can select different transmission policies. LRA also responds non-monotonically to transmission and channel conditions.

B. Complexity Analysis

  • Optimization falls from O(N^9/ϵ) grid search to O(N^3/ϵ) for general-source probability and O(N^6/ϵ) persistence; structured cases use root finding, with one O(1) case.The general-source stationary characterization avoids solving the N^3-dimensional stationary equation.Source: B. Complexity Analysis · p. 6

1) Analytical Validation:

  • The three evaluation methods agree closely, and grid-search optima coincide with analytical optima for the structured source models.This validates the system-dynamics characterization, matrix-form metric evaluation, and closed-form expressions.Source: 1) Analytical Validation · p. 7
  • For the two-state source, probability and persistence are maximized near pa = 0.477 and pa = 0.17, respectively; using the probability-optimal policy lowers persistence by 12.4%.The uniform-transition example likewise shows that optimizing one LRA metric can consume more resources while reducing the other.Source: 1) Analytical Validation · p. 7

2) Comparison with Marginal Analysis:

  • More frequent transmission can improve legitimate accuracy yet reduce LRA when it also improves Eve’s reconstruction; LRA-optimal transmission also depends on channel conditions.Improving Bob’s channel does not necessarily increase LRA monotonically, because simultaneous correct reconstruction can also become more likely.Source: 2) Comparison with Marginal Analysis · p. 7
  • A balanced marginal objective selects pa ≈ 0.8 and yields LRA ≈ 0.189, versus LRA ≈ 0.22 at pa ≈ 0.18 under direct optimization.The marginally optimized policy transmits more frequently while achieving approximately 14% lower LRA probability.Source: 2) Comparison with Marginal Analysis · p. 7, Figure 4 · p. 8
  • The balanced marginal metric can remain fixed while LRA varies with source dynamics, so weighted marginal objectives can fail to distinguish policies with different LRA probabilities.Persistent sources favor increasingly sparse transmission, whereas alternating sources favor full transmission.Source: 2) Comparison with Marginal Analysis · pp. 7–8

3) Effect of Legitimate-Channel Quality:

The channel-quality analysis shows that stronger legitimate reception can improve Bob’s accuracy without improving LRA, while the geofencing example turns an LRA threshold into spatial boundaries. The conclusion emphasizes that probability and persistence, and policies optimized for marginal objectives versus joint LRA, can favor different choices.

3) Effect of Legitimate-Channel Quality:

  • A stronger legitimate channel can initially reduce LRA probability because Bob’s accuracy gain may also increase simultaneous correct reconstruction at Eve.The resulting LRA probability first decreases and then increases as Bob’s channel success probability improves.Source: 3) Effect of Legitimate-Channel Quality · p. 8
  • Weakening Eve’s channel or accelerating source evolution can improve LRA more effectively than improving Bob’s channel when Eve’s channel is strong.The stated examples use Eve success probability 0.1 or source transition probabilities p = q = 0.9.Source: 3) Effect of Legitimate-Channel Quality · p. 8
  • The LRA-optimal transmission probability can rise and then fall with Bob’s channel quality, while preferred transmission frequency also depends on Eve’s channel and source dynamics.For p = q = 0.1 and Eve success probability 0.1, the optimal probability first increases and then decreases; with Eve success probability 0.5, the persistent source favors sparse transmission while the frequently alternating source favors full transmission at lower Bob channel quality.Source: 3) Effect of Legitimate-Channel Quality · p. 8

4) LRA-Induced Geofencing:

  • The geofencing boundary marks locations where even the optimal randomized stationary policy cannot meet the prescribed LRA-probability threshold under zero-order-hold reconstruction.The example reports a relatively large region to the left and right, where lack of blockage favors Eve’s propagation.Source: 4) LRA-Induced Geofencing · p. 9

VII. CONCLUSION

  • Greater transition asymmetry can increase LRA persistence while reducing its probability at fixed transition-probability sum, and probability and persistence can favor different policies.These conclusions are stated for two-state sources under the paper’s randomized stationary transmission and zero-order-hold reconstruction assumptions.Source: §VII · p. 9
  • The conclusions are scoped to randomized stationary transmission and prescribed zero-order-hold reconstruction; source-aware adversarial inference remains future work.The paper states that this future work may change preferred policies and resulting spatial boundaries.Source: §VII · p. 9

APPENDIX A PROOF OF LEMMA 1

The proofs construct a finite, time-homogeneous Markov process for the source and receiver states, establish its unique ergodic recurrent class, and use stationary limits to obtain the joint distribution.

APPENDIX A PROOF OF LEMMA 1

  • The induced source-and-receiver state process is Markov because its next state depends on the current state, and its transition law is time-homogeneous.The proof uses the source’s Markov property and independence from transmission and channel-state processes.Source: Appendix A · pp. 9–10

1) Uniqueness of the Recurrent Class:

  • A diagonal joint state is reachable from every state, which makes the finite induced chain’s recurrent class unique.The construction lets the source reach a target state and has both receivers update in the final slot.Source: 1) Uniqueness of the Recurrent Class · p. 10

2) Aperiodicity:

  • The source’s aperiodicity implies the unique recurrent class of the induced process is aperiodic.Source return times are also return times for a diagonal joint state when both receivers receive in the final slot.Source: 2) Aperiodicity · p. 10

APPENDIX B PROOF OF LEMMA 2

  • The induced process is ergodic on its unique recurrent class and its distribution converges from any initial state to a unique stationary distribution.The proof establishes this using finiteness, uniqueness of the recurrent class, and aperiodicity.Source: 2) Aperiodicity · p. 10, Appendix B · p. 10

APPENDIX C PROOF OF THEOREM 1

  • The stationary distribution is obtained by combining limiting source-state and joint-age distributions into the joint stationary probability.The proof first derives the limiting distributions separately, then combines them.Source: Appendix C · p. 10

1) Stationary Distribution of Joint Age Process:

The joint age distribution separates synchronized updates from cases in which either Bob or Eve has the fresher update. The stationary three-time source distribution uses the same timing cases to track the states underlying each receiver’s reconstruction.

1) Stationary Distribution of Joint Age Process:

  • The joint age process distinguishes synchronized updates, Bob-fresher states, and Eve-fresher states.For Bob-fresher states, the age gap is established at a boundary state and then persists through consecutive joint failures; the Eve-fresher case is symmetric.Source: 1) Stationary Distribution of Joint… · pp. 10–11

2) Stationary Three-Time Joint Distribution:

  • The stationary three-time joint distribution likewise separates synchronized, Bob-fresher, and Eve-fresher update times.When update times coincide, the two stored source states must match; the source then evolves from that state to its current state over the intervening steps.Source: 2) Stationary Three-Time Joint… · p. 11

3) Stationary Distribution of Induced Process:

The stationary distribution of the induced process is assembled by cases according to which receiver has the fresher update, then used to characterize LRA probability and persistence. The appendix also gives an algebraic LRA-probability expression for uniform-transition sources.

3) Stationary Distribution of Induced Process:

  • The induced process’s stationary distribution is assembled from synchronized-update, Bob-fresher, and Eve-fresher cases.The proof combines the corresponding age and three-time joint distributions to obtain the result in equation (7).Source: 3) Stationary Distribution of Induced… · p. 11

APPENDIX D PROOF OF THEOREM 2

  • The LRA probability is the stationary probability of an LRA state, while persistence is derived from probabilities that the process remains in LRA across consecutive slots.An LRA run cannot continue through a successful reception at Eve.Source: Appendix D · p. 11

APPENDIX E PROOF OF COROLLARY 4

  • For uniform-transition sources, the appendix gives an algebraic expression for long-term average LRA probability using the common probabilities of the LRA states.The expression is obtained after noting that all N(N − 1) LRA states have identical probabilities.Source: Appendix E · p. 12

APPENDIX F PROOF OF COROLLARY 5

The proof characterizes transitions within the LRA state set and gives the long-term average LRA persistence using Theorem 2.

APPENDIX F PROOF OF COROLLARY 5

  • The LRA state set contains N(N − 1) states of the form (x, x, b), where b ≠ x.Source: Appendix F · p. 12
  • A successful reception at Eve makes its reconstruction correct, so LRA transitions require Eve to receive no update.Source: Appendix F · p. 12
  • Theorem 2 gives the long-term average LRA persistence in terms of the displayed model parameters.The supplied expression includes ρ(1 − pe) and (N − 2)ηλ10.Source: Appendix F · p. 12

APPENDIX G PROOF OF COROLLARY 6

The proof analyzes the derivative of the objective through a polynomial and its coefficient signs, then treats root locations and degenerate cases to characterize maximizers.

APPENDIX G PROOF OF COROLLARY 6

  • The proof establishes β2 ≤ 0, β3 ≤ 0, and β5 ≤ 0, while β1 and β4 have parameter-dependent signs.Source: Appendix G · p. 12
  • The derivative polynomial has nonnegative discriminant after substituting the explicit coefficients, allowing its roots to be analyzed.Source: Appendix G · p. 12
  • For β1 < 0 or β1 > 0, the polynomial PF is decreasing on [0, 1], and the relevant root determines the objective's maximizing choice.The proof locates the vertex outside the interval and considers the root that may lie within it.Source: Appendix G · pp. 12–13
  • When β1β3 = 0, the proof treats affine or zero cases separately and uses endpoint comparison when F is monotone.Source: Appendix G · p. 13
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