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arXiv:1304.2444 · IEEE Transactions on Information Theory

Common Information and Secret Key Capacity

Himanshu Tyagi

Published May 20, 2013• 70 citations
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Abstract

We study the generation of a secret key of maximum rate by a pair of terminals observing correlated sources and with the means to communicate over a noiseless public communication channel. Our main result establishes a structural equivalence between the generation of a maximum rate secret key and the generation of a common randomness that renders the observations of the two terminals conditionally independent. The minimum rate of such common randomness, termed interactive common information, is related to Wyner's notion of common information, and serves to characterize the minimum rate of interactive public communication required to generate an optimum rate secret key. This characterization yields a single-letter expression for the aforementioned communication rate when the number of rounds of interaction are bounded. An application of our results shows that interaction does not reduce this rate for binary symmetric sources. Further, we provide an example for which interaction does reduce the minimum rate of communication. Also, certain invariance properties of common information quantities are established that may be of independent interest.

DOI
10.1109/tit.2013.2264355

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•Jan 5, 2016•7 citations
We study the generation of a secret key of maximum rate by a pair of terminals observing correlated sources and with the means to communicate over a noiseless public communication channel. Our main result establishes a structural equivalence between the generation of a maximum rate secret key and the generation of a common randomness that renders the observations of the two terminals conditionally independent. The minimum rate of such common randomness, termed interactive common information, is related to Wyner's notion of common information, and serves to characterize the minimum rate of interactive public communication required to generate an optimum rate secret key. This characterization yields a single-letter expression for the aforementioned communication rate when the number of rounds of interaction are bounded. An application of our results shows that interaction does not reduce this rate for binary symmetric sources. Further, we provide an example for which interaction does reduce the minimum rate of communication. Also, certain invariance properties of common information quantities are established that may be of independent interest.
We consider the generation of common randomness (CR), secret or not secret, by two user terminals with aid from a "helper" terminal. Each terminal observes a different component of a discrete memoryless multiple source. The helper aids the users by transmitting information to them over a noiseless public channel subject to a rate constraint. Furthermore, one of the users is allowed to transmit to the other user over a public channel under a similar rate constraint. We study the maximum rate of CR which can be thus generated, including under additional secrecy conditions when it must be concealed from a wiretapper. Lower bounds for the corresponding capacities are provided, and single-letter capacity formulas are obtained for several special cases of interest.
A basic two-terminal secret key generation model is considered, where the interactive communication rate between the terminals may be limited, and in particular may not be enough to achieve the maximum key rate. We first prove a multi-letter characterization of the key-communication rate region (where the number of auxiliary random variables depends on the number of rounds of the communication), and then provide an equivalent but simpler characterization in terms of concave envelopes in the case of unlimited number of rounds. Two extreme cases are given special attention. First, in the regime of very low communication rates, the key bits per interaction bit (KBIB) is expressed with a new “symmetric strong data processing constant”, which has a concave envelope characterization analogous to that of the conventional strong data processing constant. The symmetric strong data processing constant can be upper bounded by the supremum of the maximal correlation coefficient over a set of distributions, which allows us to determine the KBIB for binary symmetric sources, and conclude, in particular, that the interactive scheme is not more efficient than the one-way scheme at least in the low communication-rate regime. Second, a new characterization of the minimum interaction rate needed for achieving the maximum key rate (MIMK) is given, and we resolve a conjecture by Tyagi regarding the MIMK for (possibly nonsymmetric) binary sources. We also propose a new conjecture for binary symmetric sources that the interactive scheme is not more efficient than the one-way scheme at any communication rate.
We consider secret key (SK) generation by two terminals, each of which observes an i.i.d. source which is correlated with the other; the two sources are jointly Gaussian with known distribution. The terminals are then allowed to communicate with each other, possibly interactively in many rounds and without rate restrictions, over a noiseless two-way public channel. Randomization is permitted at each terminal. The goal is for the terminals to generate a SK comprising discrete common randomness which is concealed from an eavesdropper that has access to the public interterminal communication. We establish that this maximum rate of SK generation, i.e., SK capacity, is, as expected, the per-symbol mutual information I of the correlated sources. Our main technical contribution is a new scheme for achieving SK capacity using structured codes. In our scheme, vector quantization at rate R nats/source symbol of the source at terminal 1 is performed by employing nested lattice codes with dithering. Then, a SK is generated by the terminals from the quantized random sequence at terminal 1 and the (unquantized) Gaussian source at terminal 2, using both lossy and lossless data compression techniques. Our algorithm, using a rate-R lattice quantizer, achieves a maximum SK rate of 1/2 log 1/(e-2I+(1-e-2I)e-2R), which tends to the SK capacity I with increasing R. Thus, our result also specifies the associated tradeoff between quantization rate R and the maximum achievable SK rate. This tradeoff is shown to be optimum among a certain restricted class of schemes for SK generation that involve quantization at rate R of a Gaussian source at one terminal.
We investigate secret key generation for a ldquopair-wise independent networkrdquo model in which every pair of terminals observes correlated sources which are independent of sources observed by all other pairs of terminals. The terminals are then allowed to communicate interactively in multiple rounds over a public noiseless channel of unlimited capacity, with all such communication being observed by all the terminals. The objective is to generate a secret key shared by a given subset of terminals at the largest rate possible. All the terminals cooperate in generating the secret key, with secrecy being required from an eavesdropper which has access to the public interterminal communication. We provide a (single-letter) formula for the secrecy capacity for this model, and show a natural connection between the problem of secret key generation and the combinatorial problem of maximal packing of Steiner trees in an associated multigraph. In particular, we show that the maximum number of Steiner tree packings in the multigraph is always a lower bound for the secrecy capacity. The bound is tight for the case when all the terminals seek to share a secret key; the mentioned connection yields an explicit capacity-achieving algorithm. This algorithm, which can be executed in polynomial time, extracts a group-wide secret key of the optimum rate from the collection of optimum and mutually independent secret keys for pairs of terminals.
The basic two-terminal common randomness (CR) and key generation models are considered, where the communication between the terminals may be limited, and in particular may not be enough to achieve the maximum CR/key rate. We introduce general notions of XYXYXY-absolutely continuity and XYXYXY-concave function, and characterize the first order CR/key-communication tradeoff in terms of the evaluation of the XYXYXY-concave envelope of a functional defined on a set of distributions, which is simpler than the multi-letter characterization. Two extreme cases are given special attention. First, in the regime of very small communication rates, the CR bits per interaction bit (CRBIB) and key bits per interaction bit (KBIB) are expressed with a new symmetrical strong data processing constant, defined as the minimum of a parameter such that a certain information-theoretic functional touches its XYXYXY-concave envelope at a given source distribution. We also provide a computationally friendly strong converse bound for CRBIB and a similar (but not necessarily strong) one for KBIB in terms of the supremum of the maximal correlation coefficient over a set of distributions. The proof uses hypercontractivity and properties of the R\'enyi divergence. A criterion the tightness of the bound is given with applications to binary symmetric sources. Second, a new characterization of the minimum interaction rate needed for achieving the maximum key rate (MIMK) is given, and we resolve a conjecture by Tyagi and Narayan \cite{tyagi2013common} regarding the MIMK for binary sources. We also propose a new conjecture for binary symmetric sources.