Covert communications on Poisson packet channels

Abstract

Consider a channel where authorized transmitter Jack sends packets to authorized receiver Steve according to a Poisson process with rate λ packets per second for a time period T . Suppose that covert transmitter Alice wishes to communicate information to covert receiver Bob on the same channel without being detected by a watchful adversary Willie. We consider two scenarios. In the first scenario, we assume that warden Willie cannot look at packet contents but rather can only observe packet timings, and Alice must send information by inserting her own packets into the channel. We show that the number of packets that Alice can covertly transmit to Bob is on the order of the square root of the number of packets that Jack transmits to Steve; conversely, if Alice transmits more than that, she will be detected by Willie with high probability. In the second scenario, we assume that Willie can look at packet contents but that Alice can communicate across an M/M/1 queue to Bob by altering the timings of the packets going from Jack to Steve. First, Alice builds a codebook, with each codeword consisting of a sequence of packet timings to be employed for conveying the information associated with that codeword. However, to successfully employ this codebook, Alice must always have a packet to send at the appropriate time. Hence, leveraging our result from the first scenario, we propose a construction where Alice covertly slows down the packet stream so as to buffer packets to use during a succeeding codeword transmission phase. Using this approach, Alice can covertly and reliably transmit O(λT ) covert bits to Bob in time period T over an M/M/1 queue with service rate µ > λ.

Fundamental Limits of Covert Bit Insertion in Packets

“…In this paper, we presented the fundamental limits of a third technique for the transmission of covert information on network links, covert insertion of bits in the payload of packets with available space, where the first and second schemes were packet insertion and timing channel presented in [23], [24].…”

“…System configuration: Authorized user Jack sends packets through the network to authorized user Steve. Alice adds her information packet timings are governed by a Poisson process [23] or a renewal process [24]. If Willie can verify the sources of the packets, then Alice can transmit O(n) bits to Bob via timing if the timing channel between Alice and Bob has non-zero capacity.…”

Fundamental Limits of Covert Packet Insertion

“…Similar to the definition of covertness in [1], [2], [16]- [18], [67], [68], and invisibility in [69], [70], we define covertness: We present results under the assumption that P(H 0 ) = P(H 1 ) = 1/2. However, this results in covertness for the general case [71, Appendix A].…”

“…If we assume Alice and Bob can share a codebook and the altering of timings in the channel can be modeled by a queue, sending information via packet timing is studied for Poisson packet channels in [1,Theorem 2] and for renewal channels in [2,Theorem 5].…”

“…In this lemma, the term "packet" will refer to Jack's packets. For a fixed number of packets N , Bob scales up the IPDs by1 1−ρ 1 where 0 < ρ 1 < 1, i.e, if he receives the i th packet at τ p (i), he sends it at time τp(i) 1−ρ 1 , as shown inFig. 2.First, we show that Bob can buffer O( √ N ) packets, then we demonstrate covertness, and finally in the converse case we show that Bob cannot buffer ω( √ N ) packets covertly.…”