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An Operational Leakage Metric for Eavesdropping in Joint Sensing and Communications Systems
Nanchi Su, Xiaoye Jing, Fan Liu, George C. Alexandropoulos, Christos Masouros, Qinyu Zhang
TL;DR
Finite-record ISAC security lacks a measure of the chance that Eve meets either communication or sensing accuracy targets. This letter defines Bayesian operational leakage and separates prior knowledge from observation risk; numerically, observation-enabled risk rises with Eve’s SNR, and the conditional Gaussian approximation closely follows Bayes evaluation.
Problem
Existing information and estimation measures do not directly quantify the finite-record probability that Eve meets a communication or sensing distortion threshold.
Source: §I · p. 1Method
The paper defines operational leakage as Eve’s optimal Bayesian probability of meeting either distortion threshold and separates prior success from observation-induced risk, with a conditional Laplace approximation for evaluation.
Source: §I · p. 1Results
From −10 to 20 dB, Y rises from 0.179 to 0.513 and normalized leakage from 0.021 to 0.420; rate and CRB do not uniquely determine success, while Gaussian evaluation closely follows Bayes.
Source: §VI · pp. 5–6Takeaways & Limitations
The prior baseline captures risk from Eve’s available knowledge, while the observation-assisted probability captures the additional risk from the received signal.
Source: §VII · p. 6The conditional Gaussian approximation is suited to sufficiently concentrated posteriors; broad or multimodal posteriors still require numerical Bayes evaluation.
Source: §VII · p. 6
Abstract
from arXiv · showhide
In upcoming integrated sensing and communications (ISAC) systems, security depends on whether an eavesdropper (Eve) can recover data symbols or infer sensing information from a common observation with a legitimate receiver. However, existing information and estimation measures do not quantify this dual risk under finite observation records. This letter introduces the novel metric of operational leakage, which signifies Eve's optimal probability, in the Bayesian sense, of reconstructing communication symbols or estimating a sensing parameter within prescribed distortions. By showing that prior adversarial success probability separates the effect of side information from that of the observation, we propose a conditional Laplace approximation for efficiently evaluating the proposed metric and jointly quantifying the eavesdropping risk to the sensing and communication functionalities. Numerical results showcase that the risk enabled by adversarial observation increases with Eve's signal-to-noise ratio (SNR), conventional metrics do not uniquely determine finite-record adversarial success probability, and that the proposed conditional Gaussian approximation closely follows numerical Bayes evaluation.
I. Introduction
The introduction frames finite-record ISAC security as Eve’s chance of meeting a communication or sensing accuracy threshold, and proposes a Bayesian metric that separates prior knowledge from observation risk. The system model specifies the transmitted waveform and Eve’s observation setting.
I. Introduction
- Existing information-rate, average-exposure, and estimation-bound measures do not directly answer the finite-record probability that Eve meets either service distortion threshold.This matters when an attack succeeds upon meeting either threshold.Source: §I · p. 1
- The operational metric is Eve’s optimal Bayesian probability of meeting either a communication or sensing distortion threshold.It treats symbol reconstruction and sensing estimation jointly.Source: §I · p. 1
- A prior-success baseline and normalized observation gain separate Eve’s side-information risk from the additional risk revealed by the observation.Source: §I · p. 1
- The proposed conditional Laplace approximation uses the posterior Hessian and is assessed against numerical Bayes results for feasible designs.Source: §I · p. 1
II. System and Adversary Observation Model
- The modeled full-duplex base station transmits a composite waveform containing precoded user data and independent artificial noise across snapshots.The base station also acquires monostatic echoes, which are not used in the proposed leakage metric.Source: §II · pp. 1–2
- Eve’s channel model includes a direct base-station path and a target-reflected echo, under a narrowband, far-field, single-point-reflector setting.Other reflectors and multipath are excluded.Source: §II · p. 2
III. Adversary Side Information and Bayesian Rules
Eve’s success probability depends on what she knows before observing the signal and which Bayesian decisions she can make afterward. The model distinguishes Eve’s realized side information from public model variables and incorporates unknown quantities into her posterior.
III. Adversary Side Information and Bayesian Rules
- The leakage metric depends on Eve’s pre-observation knowledge and the Bayesian decision rules available after observing her signal.Source: §III · p. 2
A. Side Information and Latent Quantities
- Eve’s side information may include knowledge of priors for the unknown symbols, sensing parameter, and physical variables, as well as realized values such as pilots or calibration data.Source: A. Side Information and Latent… · p. 2
- Quantities unknown to Eve are assigned to the latent physical variables, while known parameters are removed from that set; public model variables remain distinct from realized side information.Source: A. Side Information and Latent… · p. 2
- The joint prior and likelihood marginalized over artificial noise define Eve’s posterior and observation-assisted success probability across independent snapshots.Source: A. Side Information and Latent… · p. 2
B. Bayesian Decision Rules
The exact leakage metric allows Eve to use measurable Bayesian rules based on her observation, side information, known model information, and private randomization, without restricting computational complexity or estimator architecture.
B. Bayesian Decision Rules
- Eve’s Bayesian rule class permits estimates based on her observation, realized side information, model information known to her, and independent private randomization.Source: B. Bayesian Decision Rules · p. 2
- The rule class imposes no computational-complexity or estimator-architecture constraint, so restricted decoder, neural, or iterative families define separate constrained leakage metrics.Source: B. Bayesian Decision Rules · p. 2
IV. Operational Joint Leakage Metric
The section defines Eve’s joint operational success through communication and sensing distortion thresholds, then evaluates the best Bayesian chance of meeting either threshold. The task losses and success event specify what counts as reconstruction or sensing success.
IV. Operational Joint Leakage Metric
- Under Gaussian signaling, communication accuracy uses squared error per complex symbol, while finite-alphabet detection can use normalized Hamming distortion with estimates restricted to the symbol alphabet.The Gaussian evaluation and numerical results use the squared-error criterion.Source: §IV · pp. 2–3
- Sensing success is defined by a task-specific error measure, such as wrapped angular error for a target direction, with separate communication and sensing thresholds.The sensing example treats angles separated by 2π as equivalent.Source: §IV · p. 3
- Eve succeeds when either distortion threshold is met, and the symbol and sensing estimates may be chosen separately rather than from one joint estimate.The paper leaves requiring success in both tasks or weighting the tasks for future work.Source: §IV · p. 3
- The OR vulnerability is the optimal probability of this joint success event under a distribution of symbols and sensing parameters, and it is convex in that distribution.For independent communication and sensing variables, separately maximizing each single-task success probability also maximizes OR success.Source: §IV · p. 3
A. Prior Baseline and Observation Assisted Risk
The section distinguishes Eve’s best success probability using side information alone from the probability after observing the signal. Operational leakage normalizes the observation-enabled increase against the remaining prior failure probability.
A. Prior Baseline and Observation Assisted Risk
- The prior success probability allows Eve to choose estimates using side information but not the observation, whereas the observation-assisted probability optimizes estimates conditional on each observation before averaging.These definitions use the distribution of the symbol and sensing parameter after averaging over unknown physical variables.Source: A. Prior Baseline and Observation… · p. 3
- Operational leakage is the observation-induced increase in adversarial success probability divided by the remaining chance of failure using side information alone.A value of zero means the observation adds no advantage; if prior success is one, the normalized value is undefined and prior success should be reported.Source: A. Prior Baseline and Observation… · p. 3
- Both success probabilities are nondecreasing in either distortion threshold, and the Bayesian definition can use observation models incorporating multipath, multiple targets, or wideband signaling.A less informative observation cannot increase adversarial success probability.Source: A. Prior Baseline and Observation… · p. 3
V. Conditional Gaussian Evaluation
Because the exact posterior characterization is difficult to express in closed form, the section proposes a conditional Laplace approximation that preserves the order of Eve’s decision and observation averaging.
V. Conditional Gaussian Evaluation
- The exact posterior characterization in (17) can hardly have a closed form, motivating a conditional Laplace approximation.The approximation first optimizes posterior success for each observation, then averages over observations.Source: §V · p. 3
- The approximation uses squared error for communication reconstruction under Gaussian signaling.This is the communication criterion used by the Gaussian evaluation.Source: §V · p. 3
- Finite-alphabet detection requires posterior sums over discrete symbol hypotheses, potentially combined with integration over continuous sensing and nuisance parameters; the Gaussian formulas do not directly evaluate symbol or coded-message decoding success.Source: §V · p. 3
A. Conditional Laplace Posterior
The conditional Laplace posterior is built around the MAP point and observed posterior Hessian. This local approximation differs from observation-independent Fisher-information criteria and has stated scope limits.
A. Conditional Laplace Posterior
- The MAP point and observed posterior Hessian define the local quantities used by the conditional approximation.The Hessian is evaluated at the MAP point.Source: A. Conditional Laplace Posterior · p. 4
- The conditional approximation uses the observed posterior Hessian in (23), consistent with a Laplace posterior approximation, rather than observation-independent Fisher information or CRB criteria.Source: A. Conditional Laplace Posterior · p. 4
- Singular or multimodal cases are not rigorously covered by the local approximation, and ill-conditioned H_νν may be regularized with εI.Source: A. Conditional Laplace Posterior · p. 4
- Removing all unknown physical variables ν at once retains posterior covariance between the sensing variable and the real and imaginary symbol components.Source: A. Conditional Laplace Posterior · p. 4
B. Conditional OR Success Probability
The conditional approximation mirrors Eve’s decision process: optimize OR-success under the posterior for each observation, then average; a prior-only calculation provides the baseline.
B. Conditional OR Success Probability
- For each observation, Eve’s estimates are optimized for OR-success under the Gaussian posterior, and the optimized probability is then averaged over observations.Posterior-mean or MAP plug-in estimates are cheaper but need not maximize OR-success.Source: B. Conditional OR Success Probability · p. 4
- The prior success probability uses the same success calculation on Eve’s prior, separating side information from the observation’s additional risk.The prior may be exact, empirical, or Laplace-approximated; separately approximated prior and conditional distributions need not preserve their expected ordering.Source: B. Conditional OR Success Probability · p. 4
- Under the Gaussian approximation, joint sensing-and-communication success must use joint covariance or joint samples, not the product of individual success probabilities.The success probability is for meeting either task’s threshold, so the overlap between the two success events matters.Source: B. Conditional OR Success Probability · p. 4
- The union bound min{1, eP_s + eP_c} is only a rough Gaussian-approximation sanity check, not a bound on P★.The approximation can be diagnostically normalized only when 0 ≤ eP_0 ≤ eP_Y ≤ 1; otherwise report the pair without normalization.Source: B. Conditional OR Success Probability · pp. 4–5
VI. Numerical Results
Across feasible designs, the conditional Gaussian approximation closely tracks numerical Bayesian evaluation, while conventional metrics do not uniquely rank adversarial success. Eve’s observation and looser distortion thresholds are associated with higher success probabilities.
VI. Numerical Results
- The validation uses 200 feasible (F, Z) candidate pairs under the same power, communication, and sensing constraints, rather than optimizing the transmit design.The setup uses 8 base-station transmit antennas, 4 Eve receive antennas, and 100 snapshots at nominal thresholds δc = 0.10 and δL = 1°.Source: §VI · p. 5
- P★_0 = 0.161 stays unchanged with Eve SNR, while P★_Y rises from 0.179 at −10 dB to 0.513 at 20 dB and normalized leakage rises from 0.021 to 0.420.The observation therefore adds an increasing increment in Eve’s success beyond the prior baseline.Source: §VI · p. 5
- For secrecy rate, (ρS, τK) = (0.181, 0.120), and for sensing CRB, (ρS, τK) = (0.048, 0.032), indicating weak ranking consistency with adversarial success.These rate- and estimation-based quantities cannot uniquely determine finite-record adversarial success probability.Source: §VI · p. 5
- ρS = 0.977 and MAE = 0.021 show that the conditional Gaussian approximation preserves candidate ordering and closely follows numerical Bayesian evaluation.The MAE is the average probability error across N = 200 feasible candidate designs.Source: §VI · p. 6
- Increasing either communication or sensing distortion threshold enlarges the success region and raises P★_Y.The two candidates have similar secrecy rates and sensing CRBs yet differ in adversarial success across the considered thresholds.Source: §VI · p. 6
VII. Conclusions
The conclusion separates Eve’s prior-knowledge risk from the additional risk revealed by the received signal. It also qualifies the conditional Gaussian approximation and notes that conventional metrics do not uniquely determine finite-record success probability.
VII. Conclusions
- The prior baseline captures Eve’s available-knowledge risk, while the observation-assisted probability captures additional risk from the received signal.Source: §VII · p. 6
- The conditional Gaussian approximation efficiently evaluates adversarial success when the posterior is sufficiently concentrated; broad or multimodal posteriors still require numerical Bayes evaluation.Source: §VII · p. 6
- Secrecy-rate and CRB measures do not uniquely determine finite-record adversarial success probability, and the prior/observation risk split depends on distortion thresholds.Source: §VII · p. 6