Fundamental Limits of Covert Packet Insertion

Covert communication conceals the existence of the transmission from a watchful adversary. We consider the fundamental limits for covert communications via packet insertion over packet channels whose packet timings are governed by a renewal process of rate <inline-formula> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula>. Authorized transmitter Jack sends packets to authorized receiver Steve, and covert transmitter Alice wishes to transmit packets to covert receiver Bob without being detected by watchful adversaries Willie 1 and Willie 2. Willies cannot authenticate the source of the packets or collaborate. Hence, each Willie looks for statistical anomalies in the packet stream from Jack to Steve to attempt detection of unauthorized packet insertion. First, we consider a special case where the packet timings are governed by a Poisson process and we show that Alice can covertly insert <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(\sqrt {\lambda T})$ </tex-math></inline-formula> packets for Bob in a time interval of length <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>; conversely, if Alice inserts <inline-formula> <tex-math notation="LaTeX">$\omega (\sqrt {\lambda T})$ </tex-math></inline-formula> packets, she will be detected by Willie 1 or Willie 2 with high probability. Then, we extend our results to general renewal channels and show that in a stream of <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> packets transmitted by Jack, Alice can covertly insert <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(\sqrt {N})$ </tex-math></inline-formula> packets; if she inserts <inline-formula> <tex-math notation="LaTeX">$\omega (\sqrt {N})$ </tex-math></inline-formula> packets, she will be detected by Willie 1 or Willie 2, with high probability.

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