[Paper Review] Covert Communication Gains from Adversary's Ignorance of Transmission Time

Boulat A. Bash, Dennis Goeckel+1 more|arXiv (Cornell University)|Mar 5, 2014

Wireless Communication Security Techniques31 references3 citations

TL;DR

This paper demonstrates that when an adversary (Willie) is ignorant of the transmission time, Alice can covertly communicate significantly more data to Bob than the standard square root law allows. By transmitting in a single, secret slot among T(n) slots, Alice achieves a covert throughput of Θ(min{√(n log T(n)), n}) bits, leveraging Willie’s uncertainty to amplify covert capacity beyond the √n limit of traditional models.

ABSTRACT

The recent square root law (SRL) for covert communication demonstrates that Alice can reliably transmit $\mathcal{O}(\sqrt{n})$ bits to Bob in $n$ uses of an additive white Gaussian noise (AWGN) channel while keeping ineffective any detector employed by the adversary; conversely, exceeding this limit either results in detection by the adversary with high probability or non-zero decoding error probability at Bob. This SRL is under the assumption that the adversary knows when Alice transmits (if she transmits); however, in many operational scenarios he does not know this. Hence, here we study the impact of the adversary's ignorance of the time of the communication attempt. We employ a slotted AWGN channel model with $T(n)$ slots each containing $n$ symbol periods, where Alice may use a single slot out of $T(n)$. Provided that Alice's slot selection is secret, the adversary needs to monitor all $T(n)$ slots for possible transmission. We show that this allows Alice to reliably transmit $\mathcal{O}(\min\{\sqrt{n\log T(n)},n\})$ bits to Bob (but no more) while keeping the adversary's detector ineffective. To achieve this gain over SRL, Bob does not have to know the time of transmission provided $T(n)<2^{c_{ m T}n}$, $c_{ m T}=\mathcal{O}(1)$.

Motivation & Objective

• To investigate how Willie’s ignorance of transmission time affects covert communication limits in noisy channels.
• To model practical scenarios where Alice transmits a short message within a longer time window, unknown to the adversary.
• To determine the maximum number of covert bits Alice can reliably send while evading detection.
• To analyze the trade-off between transmission duration, slot selection secrecy, and detection probability.

Proposed method

• Model a slotted AWGN channel with T(n) slots, each containing n symbol periods, where Alice uses only one secret slot for transmission.
• Assume Willie monitors all T(n) slots but does not know which one Alice uses, increasing his detection difficulty.
• Use a random coding argument with binary modulation to construct codebooks that minimize Bob’s decoding error while keeping Willie’s detection probability low.
• Apply the square root law (SRL) to the effective channel length, adjusting for the fact that Willie must monitor T(n) slots instead of just one.
• Derive bounds on the false alarm and missed detection probabilities at Willie’s detector, showing they can be made arbitrarily small.
• Use the union bound and Gaussian tail bounds to analyze Bob’s decoding error probability under peak power constraints.

Experimental results

Research questions

• RQ1 Can Alice increase her covert throughput by exploiting the adversary’s ignorance of transmission time?
• RQ2 What is the fundamental limit on the number of covert bits Alice can send when Willie does not know when transmission occurs?
• RQ3 How does the number of available time slots T(n) affect the achievable covert communication rate?
• RQ4 Can Bob decode reliably even without knowing the transmission time, provided T(n) grows sub-exponentially with n?
• RQ5 Does the adversary’s uncertainty about transmission time allow Alice to exceed the standard √n covert capacity limit?

Key findings

• Alice can reliably transmit Θ(min{√(n log T(n)), n}) covert bits to Bob when Willie is ignorant of the transmission time.
• The covert capacity gain scales as √(n log T(n)) when T(n) grows sub-exponentially with n, i.e., T(n) < 2^{c_T n} for some constant c_T.
• The improvement over the standard square root law (which caps at √n bits) arises from Willie’s need to monitor T(n) slots, increasing his uncertainty.
• Bob does not need to know the transmission time in advance as long as T(n) grows sub-exponentially, preserving the covert gain.
• The result holds even under peak power constraints and remains robust when Willie’s channel noise is known or unknown.
• Exceeding ω(√(n log T(n))) bits leads to either detection by Willie with high probability or non-zero decoding error at Bob in the asymptotic limit.