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IEEE International Symposium on Information Theory (ISIT)

Secret key generation for correlated Gaussian sources

Sirin Nitinawarat

Published Jul 1, 2008• 24 citations
Publisher

Abstract

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.

DOI
10.1109/isit.2008.4595077

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arXiv:0906.1835•IEEE Transactions on Information Theory•Feb 1, 2012•55 citations
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.
Secret key generation by multiple terminals is considered based on their observations of jointly distributed Gaussian signals, followed by public communication among themselves. Exploiting an inherent connection between secrecy generation and lossy data compression, two main contributions are made. The first is a characterization of strong secret key capacity, and entails a converse proof technique that is valid for real-valued (and not necessarily Gaussian) as well as finite-valued signals. The capacity formula acquires a simple form when the terminals observe “symmetrically correlated” jointly Gaussian signals. For the latter setup with two terminals, considering schemes that involve quantization at one terminal, the best rate of an achievable secret key is characterized as a function of quantization rate; secret key capacity is attained as the quantization rate tends to infinity. Structured codes are shown to attain the optimum tradeoff between secret key rate and quantization rate, constituting our second main contribution.
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 problem of simultaneously generating a secret key (SK) and private key (PK) pair among three terminals via public discussion is investigated, in which each terminal observes a component of correlated sources. All three terminals are required to generate a common secret key concealed from an eavesdropper that has access to public discussion, while two designated terminals are required to generate an extra private key concealed from both the eavesdropper and the remaining terminal. An outer bound on the SK-PK capacity region was established in [1], and was shown to be achievable for one case. In this paper, achievable schemes are designed to achieve the outer bound for the remaining two cases, and hence the SK-PK capacity region is established in general. The main technique lies in the novel design of a random binning-joint decoding scheme that achieves the existing outer bound.
Abstract—We study the secret-key capacity in a joint source-channel coding setup—the terminals are connected over a discrete memoryless channel and have access to side information, modelled as a pair of discrete memoryless source sequences. As our main result, we establish the upper and lower bounds on the secret-key capacity. In the lower bound expression, the equivocation terms of the source and channel components are functionally additive even though the coding scheme generates a single secret-key by jointly taking into account the source and channel equivocations. Our bounds coincide, thus establishing the capacity, when the underlying wiretap channel can be decomposed into a set of independent, parallel, and reversely degraded channels. For the case of parallel Gaussian channels and jointly Gaussian sources we show that Gaussian codebooks achieve the secret-key capacity. In addition, when the eavesdropper also observes a correlated side information sequence, we establish the secret-key capacity when both the source and channel of the eavesdropper are a degraded version of the legitimate receiver. We finally also treat the case when a public discussion channel is available, propose a separation based coding scheme, and establish its optimality when the channel output symbols of the legitimate receiver and eavesdropper are conditionally independent given the input. Index Terms—Information theoretic security, joint sourcechannel coding, public discussion, secret-key agreement, wiretap channel. I.
The problem of simultaneously generating a secret key (SK) and private key (PK) pair among three terminals via public discussion is investigated. In this problem, each terminal observes a component of correlated sources. All three terminals are required to generate the common SK to be concealed from an eavesdropper that has access to the public discussion, while two designated terminals are required to generate an extra PK to be concealed from both the eavesdropper and the remaining terminal. An outer bound on the SK-PK capacity region was established by Ye and Narayan, and was shown to be achievable for a special case. In this paper, the SK-PK capacity region is established in general by developing schemes to achieve the outer bound for the remaining two cases. The main technique lies in the novel design of a random binning-joint decoding scheme that achieves the existing outer bound.
We study the secret-key capacity in a joint source-channel coding setup-the terminals are connected over a discrete memoryless channel and have access to side information, modelled as a pair of discrete memoryless source sequences. As our main result, we establish the upper and lower bounds on the secret-key capacity. In the lower bound expression, the equivocation terms of the source and channel components are functionally additive even though the coding scheme generates a single secret-key by jointly taking into account the source and channel equivocations. Our bounds coincide, thus establishing the capacity, when the underlying wiretap channel can be decomposed into a set of independent, parallel, and reversely degraded channels. For the case of parallel Gaussian channels and jointly Gaussian sources we show that Gaussian codebooks achieve the secret-key capacity. In addition, when the eavesdropper also observes a correlated side information sequence, we establish the secret-key capacity when both the source and channel of the eavesdropper are a degraded version of the legitimate receiver. We finally also treat the case when a public discussion channel is available, propose a separation based coding scheme, and establish its optimality when the channel output symbols of the legitimate receiver and eavesdropper are conditionally independent given the input.