  A novel private communication framework is proposed where privacy is induced
by transmitting over a channel instances of linear inverse problems that are
identifiable to the legitimate receiver but unidentifiable to an eavesdropper.
The gap in identifiability is created in the framework by leveraging secret
knowledge between the transmitter and the legitimate receiver. Specifically,
the case where the legitimate receiver harnesses a secret block structure to
decode a transmitted block-sparse message from underdetermined linear
measurements in conditions where classical compressed sensing would provably
fail is examined. The applicability of the proposed scheme to practical
multiple-access wireless communication systems is discussed. The protocol's
privacy is studied under a single transmission, and under multiple
transmissions without refreshing the secret block structure. It is shown that,
under a specific scaling of the channel dimensions and transmission parameters,
the eavesdropper can attempt to overhear the block structure from the
fourth-order moments of the channel output. Computation of a statistical lower
bound suggests that the proposed fourth-order moment secret block estimation
strategy is near optimal. The performance of a spectral clustering algorithm is
studied to that end, defining scaling laws on the lifespan of the secret key
before the communication is compromised. Finally, numerical experiments
corroborating the theoretical findings are conducted.
