Correction to "Covert Communication Over Noisy Channels"

This is a small correction to an erroneous claim in [1]. Many thanks to Meng-Che Chang for pointing out the error.

Erroneous statement

The erroneous statement concerns the scaling allowed for $\omega_n$ in Theorem 1 and Theorem 2. Therein, I wrote that $\omega_n\in o(1)\cap \omega(\frac{1}{\sqrt{n}})$, which effectively says that $\omega_n$ vanishes for large $n$ but does not decay too fast. Unfortunately, requiring a decay slower than $\frac{1}{\sqrt{n}}$ is not quite enough because I later on require quantities of the form $n\exp(-\omega_n\sqrt{n})$ to vanish, see, e.g, Eq. (41) and Eq. (79).

Fixing the error

This error can be corrected by requiring $\omega_n\in o(1)\cap \omega(\frac{\log(n^{1+\gamma})}{\sqrt{n}})$ for any $\gamma>0$.

References

1. M. R. Bloch, “Covert Communication over Noisy Channels: A Resolvability Perspective,” IEEE Transactions on Information Theory, vol. 62, no. 5, pp. 2334–2354, May 2016.

   We consider the situation in which a transmitter attempts to communicate reliably over a discrete memoryless channel, while simultaneously ensuring covertness (low probability of detection) with respect to a warden, who observes the signals through another discrete memoryless channel. We develop a coding scheme based on the principle of channel resolvability, which generalizes and extends prior work in several directions. First, it shows that irrespective of the quality of the channels, it is possible to communicate on the order of \sqrt n reliable and covert bits over n channel uses if the transmitter and the receiver share on the order of \sqrt n key bits. This improves upon earlier results requiring on the order of \sqrt n\log n key bits. Second, it proves that if the receiver’s channel is better than the warden’s channel in a sense that we make precise, it is possible to communicate on the order of \sqrt n reliable and covert bits over n channel uses without a secret key. This generalizes earlier results established for binary symmetric channels. We also identify the fundamental limits of covert and secret communications in terms of the optimal asymptotic scaling of the message size and key size, and we extend the analysis to Gaussian channels. The main technical problem that we address is how to develop concentration inequalities for low-weight sequences. The crux of our approach is to define suitably modified typical sets that are amenable to concentration inequalities.t

   @article{Bloch2015b, author = {Bloch, Matthieu R.}, journal = {IEEE Transactions on Information Theory}, title = {Covert Communication over Noisy Channels: A Resolvability Perspective}, year = {2016}, issn = {0018-9448}, month = may, number = {5}, pages = {2334-2354}, volume = {62}, creationdate = {2015-03-31T00:00:00}, doi = {10.1109/TIT.2016.2530089}, eprint = {1503.08778}, file = {:2016-Bloch-IEEETransIT.pdf:PDF}, groups = {Steganography and covert communications}, keywords = {AWGN channels;Encoding;Memoryless systems;Noise measurement;Reliability theory;Covert communications;Shannon theory;low probability of detection;physical-layer security}, owner = {mattbloch} }