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IEEE International Symposium on Information Theory (ISIT)

Agreement of a restricted secret key

Chung Chan

Published Jul 1, 2012• 10 citations
Publisher

Abstract

A secret key agreement problem is proposed with the additional restriction that the key is a function of a given secret source. An inner bound to the achievable rates, error and secrecy exponents is derived. The maximum key rate strongly achievable with positive exponents is characterized. The result leads to class of admissible restrictions on the key functions that does not diminish the strongly achievable key rate.

DOI
10.1109/isit.2012.6283585

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A secret key agreement problem is proposed with the additional restriction that the key is a function of a given secret source. An inner bound to the achievable rates, error and secrecy exponents is derived. The maximum key rate strongly achievable with positive exponents is characterized. The result leads to class of admissible restrictions on the key functions that does not diminish the strongly achievable key rate.
For the multiterminal secret key agreement problem, new single-letter lower bounds are obtained on the minimum public discussion rate required to achieve any given secret key rate below the secrecy capacity. The results apply to the general source model without helpers or wiretapper's side information, but can be strengthened for hypergraphical sources. In particular, for the pairwise independent network, our results yield a complete characterization of the maximum secret key rate achievable under a constraint on the total discussion rate.
The multiterminal secret key agreement problem by public discussion is formulated with an additional source compression step where, prior to the public discussion phase, users independently compress their private sources to filter out strongly correlated components in order to generate a common secret key. The objective is to maximize the achievable key rate as a function of the joint entropy of the compressed sources. Since the maximum achievable key rate captures the total amount of information mutual to the compressed sources, an optimal compression scheme essentially maximizes the multivariate mutual information per bit of randomness of the private sources, and can therefore be viewed more generally as a dimension reduction technique. Single-letter lower and upper bounds on the maximum achievable key rate are derived for the general source model, and an explicit polynomial-time computable formula is obtained for the pairwise independent network model. In particular, the converse results and the upper bounds are obtained from those of the related secret key agreement problem with rate-limited discussion. A precise duality is shown for the two-user case with one-way discussion, and such duality is extended to obtain the desired converse results in the multi-user case. In addition to posing new challenges in information processing and dimension reduction, the compressed secret key agreement problem helps shed new light on resolving the difficult problem of secret key agreement with rate-limited discussion by offering a more structured achieving scheme and some simpler conjectures to prove.
For the multiterminal secret key agreement problem under a private source model, it is known that the maximum key rate, i.e., the secrecy capacity, can be achieved through communication for omniscience, but the omniscience strategy can be strictly suboptimal in terms of minimizing the public discussion rate. While a single-letter characterization is not known for the minimum discussion rate needed for achieving the secrecy capacity, we derive single-letter lower bounds that yield some simple conditions for omniscience to be discussion-rate optimal. These conditions turn out to be enough to deduce the optimality of omniscience for a large class of sources, including the hypergraphical sources. We also extend our results to more general class of multiterminal sources with helpers and silent users.
We consider the secret key generation problem when sources are randomly excited by the sender and there is a noiseless public discussion channel. Our setting is thus similar to recent works on channels with action-dependent states, where the channel state may be influenced by some of the parties involved. We derive single-letter expressions for the secret key capacity through a type of source emulation analysis. We also derive lower bounds on the achievable reliability and secrecy exponents, i.e., the exponential rates of decay of the probability of decoding error and of the information leakage. These exponents allow us to determine a set of strongly achievable secret key rates. For degraded eavesdroppers, the maximum strongly achievable rate equals the secret key capacity; our exponents can also be specialized to previously known results. In deriving our strong achievability results, we introduce a coding scheme that combines wiretap coding (to excite the channel) and key extraction (to distill keys from residual randomness). The secret key capacity is naturally seen to be a combination of both source- and channel-type randomness. Through examples, we illustrate a fundamental interplay between the portion of the secret key rate due to each type of randomness. We also illustrate inherent tradeoffs between the achievable reliability and secrecy exponents. Our new scheme also naturally accommodates rate limits on the public discussion. We show that under rate constraints, we are able to achieve larger rates than those that can be attained through a pure source emulation strategy.
A mutual dependence expression is established for the secret key agreement problem when all users are active. In certain source networks, the expression can be interpreted as certain notions of connectivity and network information flow. In particular, the secrecy problem can be mapped to a new class of network coding problems with selectable links and undirected broadcast links. For such networks, the secrecy capacities serve as upper bounds on the maximum network throughputs, while the network coding solutions can be used for secret key agreement.