Covert Communication Gains from Adversary's Ignorance of Transmission Time

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 $T(n)<2^{c_{\rm T}n}$, $c_{\rm T}=\mathcal{O}(1)$.Comment: v2: updated references/discussion of steganography, no change in results; v3: significant update, includes new theorem 1.2; v4 and v5: fixed minor technical issue