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arXiv:1305.4314 · IEEE International Symposium on Information Theory (ISIT)

Secure Cascade Channel Synthesis

Sanket Satpathy, Paul Cuff

Published Jul 1, 2013• 25 citations
arXivHTMLPDF

Abstract

We consider the problem of generating correlated random variables in a distributed fashion, where communication is constrained to a cascade network. The first node in the cascade observes an i.i.d. sequence$X^{n}$locally before initiating communication along the cascade. All nodes share bits of common randomness that are independent of$X^{n}$. We consider secure synthesis—random variables produced by the system appear to be appropriately correlated and i.i.d. even to an eavesdropper who is cognizant of the communication transmissions. We characterize the optimal tradeoff between the amount of common randomness used and the required rates of communication. We find that not only does common randomness help, its usage exceeds the communication rate requirements. The most efficient scheme is based on a superposition codebook, with the first node selecting messages for all downstream nodes. We also provide a fleeting view of related problems, demonstrating how the optimal rate region may shrink or expand.

DOI
10.1109/isit.2013.6620767

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Exploiting correlated observations at multiple remote nodes for generating a secret common randomness to facilitate secure communication of data between the terminals has been addressed by many authors [1][2][3]. But in many applications such as sensor networks, the correlated observations might themselves be the data that need to be communicated. We address this problem of secure distributed source coding where we want to leak the least amount of information about the sources to an eavesdropper who may have access to a correlated observation of its own and who can listen in on the communication between the distributed encoders. The focus here is on minimizing the amount of information revealed to the eavesdropper rather than on minimizing the rate required to effect the communication.
We study the problem of synthesising a two-user broadcast channel using a common message, where each output terminal shares an independent source of randomness with the input terminal. This generalises two problems studied in the literature (Cuff, IEEE Trans. Inform. Theory, 2013; Kurri et.al., IEEE Trans. Inform. Theory, 2021). We give an inner bound on the tradeoff region between the rates of communication and shared randomness, and a lower bound on the minimum communication rate. Although the bounds presented here are not tight in general, they are tight for some special cases, including the aforementioned problems.
We investigate the problem of secure communication in a simple network with three communicating parties, two distributed sources who communicate over orthogonal channels to one destination node. The cooperation between the sources is restricted to a rate limited common random source they both observe. The communication channels are erasure channels with strictly causal channel state information of the destination available publicly. A passive adversary is present in the system eavesdropping on any one of the channels. We design a linear scheme that ensures secrecy against the eavesdropper. By deriving an outer bound for the problem we prove that the scheme is optimal in certain special cases.
This dissertation concerns the secure processing of distributed data by multi-ple terminals, using interactive public communication among themselves, in order to accomplish a given computational task. In the setting of a probabilistic multiterminal source model in which several terminals observe correlated random signals, we analyze secure distributed data processing protocols that harness the correlation in the data. The specific tasks considered are: computing functions of the data under secrecy requirements; generating secretly shared bits with minimal rate of public communication; and securely sharing bits in presence of a querying eavesdropper. In studying these various secure distributed processing tasks, we adopt a unified approach that entails examining the form of underlying common randomness (CR) that is generated at the terminals during distributed processing. We make the case that the exact form of established CR is linked inherently to the data processing task at hand, and its characterization can lead to a structural understanding of the associated algorithms. An identification of the underlying CR and its decomposition into independent components, each with a different operational significance, is
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We study common randomness where two parties have access to i.i.d. samples from a known random source, and wish to generate a shared random key using limited (or no) communication with the largest possible probability of agreement. This problem is at the core of secret key generation in cryptography, with connections to communication under uncertainty and locality sensitive hashing. We take the approach of treating correlated sources as a critical resource, and ask whether common randomness can be generated resource-efficiently. We consider two notable sources in this setup arising from correlated bits and correlated Gaussians. We design the first explicit schemes that use only a polynomial number of samples (in the key length) so that the players can generate shared keys that agree with constant probability using optimal communication. The best previously known schemes were both non-constructive and used an exponential number of samples. In the amortized setting, we characterize the largest achievable ratio of key length to communication in terms of the external and internal information costs, two well-studied quantities in theoretical computer science. In the relaxed setting where the two parties merely wish to improve the correlation between the generated keys of length k, we show that there are no interactive protocols using o(k) bits of communication having agreement probability even as small as 2–o(k). For the related communication problem where the players wish to compute a joint function f of their inputs using i.i.d samples from a known source, we give a simultaneous message passing protocol using 2O(c) bits where c is the interactive randomized public-coin communication complexity of f. This matches the lower bound shown previously while the best previously known upper bound was doubly exponential in c. Our schemes reveal a new connection between common randomness and unbiased error-correcting codes, e.g., dual-BCH codes and their analogues in Euclidean space.