Guaranteed Private Communication with Secret Block Structure

TL;DR

This paper introduces a private communication scheme utilizing secret block structures in linear inverse problems, ensuring privacy against eavesdroppers while enabling legitimate decoding even in underdetermined systems, with practical implications for wireless communications.

Contribution

The paper presents a novel privacy framework based on secret block structures that allows secure decoding in underdetermined linear systems, extending compressed sensing techniques with theoretical and practical insights.

Findings

Eavesdropper's ability to infer block structure from fourth-order moments under certain conditions.

Spectral clustering algorithm's effectiveness in estimating secret block structures.

Numerical experiments validate theoretical privacy and decoding performance.

Abstract

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.