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 singleletter 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. Index Terms-Common information, common randomness, interactive communication, interactive common information, secret key capacity.
It was recently shown that estimating the Shannon entropy H(p) of a discrete k-symbol distribution p requires Θ(k/ log k) samples, a number that grows nearlinearly in the support size. In many applications H(p) can be replaced by the more general Rényi entropy of order α, H α (p). We determine the number of samples needed to estimate H α (p) for all α, showing that α < 1 requires super-linear, roughly k 1/α samples, noninteger α > 1 requires near-linear, roughly k samples, but integer α > 1 requires only Θ(k 1−1/α ) samples.In particular, estimating H 2 (p), which arises in security, DNA reconstruction, closeness testing, and other applications, requires only Θ( √ k) samples. The estimators achieving these bounds are simple and run in time linear in the number of samples. *
Abstract-A subset of a set of terminals that observe correlated signals seek to compute a function of the signals using public communication. It is required that the value of the function be concealed from an eavesdropper with access to the communication. We show that the function is securely computable if and only if its entropy is less than the capacity of a new secrecy generation model, for which a single-letter characterization is provided.
We consider information theoretic secret key agreement and secure function computation by multiple parties observing correlated data, with access to an interactive public communication channel. Our main result is an upper bound on the secret key length, which is derived using a reduction of binary hypothesis testing to multiparty secret key agreement. Building on this basic result, we derive new converses for multiparty secret key agreement. Furthermore, we derive converse results for the oblivious transfer problem and the bit commitment problem by relating them to secret key agreement. Finally, we derive a necessary condition for the feasibility of secure computation by trusted parties that seek to compute a function of their collective data, using an interactive public communication that by itself does not give away the value of the function. In many cases, we strengthen and improve upon previously known converse bounds. Our results are single-shot and use only the given joint distribution of the correlated observations. For the case when the correlated observations consist of independent and identically distributed (in time) sequences, we derive strong versions of previously known converses.
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