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Distributed Hypothesis Testing Under A Covertness Constraint

Ismaila Salihou Adamou, Michèle Wigger

TL;DR

The paper advances distributed hypothesis testing by introducing a non-alert covertness constraint from a warden and deriving the covert Stein-exponent for sublinear-rate sensor-to-decision-center communication over DMCs. It shows a channel-connectivity dichotomy: partially-connected DMCs achieve the noiseless sublinear exponent of Shalaby and Papamarcou, independent of the DMC details, while fully-connected DMCs allow achievability gains over the local exponent without requiring a secret key, with divergence-based covertness decaying essentially exponentially in $n$. The exponents are defined via type-based optimizations $E_1$, $E_2$, $E_3$, and the results hold independently of the Type-I error threshold $\epsilon$. The paper also extends prior bounds to arbitrary secret-key lengths and provides constructive achievability schemes, with future work including a full converse for fully-connected channels and covertness under both hypotheses.

Abstract

We study distributed hypothesis testing under a covertness constraint in the non-alert situation, which requires that under the null-hypothesis an external warden be unable to detect whether communication between the sensor and the decision center is taking place. We characterize the achievable Stein exponent of this setup when the channel from the sensor to the decision center is a partially-connected discrete memoryless channel (DMC), i.e., when certain output symbols can only be induced by some of the inputs. The Stein-exponent in this case, does not depend on the specific transition law of the DMC and equals Shalaby and Papamarcou's exponent without a warden but where the sensor can send $k$ noise-free bits to the decision center, for $k$ a function that is sublinear in the observation length $n$. For fully-connected DMCs, we propose an achievable Stein-exponent and show that it can improve over the local exponent at the decision center. All our coding schemes do not require that the sensor and decision center share a common secret key, as commonly assumed in covert communication. Moreover, in our schemes the divergence covertness constraint vanishes (almost) exponentially fast in the obervation length $n$, again, an atypical behaviour for covert communication.

Distributed Hypothesis Testing Under A Covertness Constraint

TL;DR

The paper advances distributed hypothesis testing by introducing a non-alert covertness constraint from a warden and deriving the covert Stein-exponent for sublinear-rate sensor-to-decision-center communication over DMCs. It shows a channel-connectivity dichotomy: partially-connected DMCs achieve the noiseless sublinear exponent of Shalaby and Papamarcou, independent of the DMC details, while fully-connected DMCs allow achievability gains over the local exponent without requiring a secret key, with divergence-based covertness decaying essentially exponentially in . The exponents are defined via type-based optimizations , , , and the results hold independently of the Type-I error threshold . The paper also extends prior bounds to arbitrary secret-key lengths and provides constructive achievability schemes, with future work including a full converse for fully-connected channels and covertness under both hypotheses.

Abstract

We study distributed hypothesis testing under a covertness constraint in the non-alert situation, which requires that under the null-hypothesis an external warden be unable to detect whether communication between the sensor and the decision center is taking place. We characterize the achievable Stein exponent of this setup when the channel from the sensor to the decision center is a partially-connected discrete memoryless channel (DMC), i.e., when certain output symbols can only be induced by some of the inputs. The Stein-exponent in this case, does not depend on the specific transition law of the DMC and equals Shalaby and Papamarcou's exponent without a warden but where the sensor can send noise-free bits to the decision center, for a function that is sublinear in the observation length . For fully-connected DMCs, we propose an achievable Stein-exponent and show that it can improve over the local exponent at the decision center. All our coding schemes do not require that the sensor and decision center share a common secret key, as commonly assumed in covert communication. Moreover, in our schemes the divergence covertness constraint vanishes (almost) exponentially fast in the obervation length , again, an atypical behaviour for covert communication.
Paper Structure (11 sections, 4 theorems, 36 equations, 1 figure)

This paper contains 11 sections, 4 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Results
  4. Proof of achievability results
  5. Achievability of \ref{['eq:Ia']}
  6. Achievability result in \ref{['eq:Ib']}
  7. Converse result in \ref{['eq:Ia']}
  8. Conclusion
  9. Proof of Lemma \ref{['lem:divergence']}
  10. Proof of Lemma \ref{['lem:lemma_low_weight']}
  11. Proof of Lemma \ref{['lem:lemma_ps']}

Key Result

Theorem 1

Fix $\epsilon \in [0,1)$. Our achievability results do not require that the sensor and decision center use the shared secret key $S$. The covertness constraint $d_n$ can be made to vanish exponentially fast in $n$ for the result in eq:Ib and arbitrary close to exponentially fast in $n$ for the achievability result in eq:Ia.

Figures (1)

  • Figure 1: Distributed hypothesis testing with an external warden.

Theorems & Definitions (8)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Remark 2: Improvement over the Local Test
  • Example 1
  • Lemma 1
  • Lemma 2: Covertness Constraint Implies Low-Weight Inputs
  • Lemma 3