Common Information Duality in Relation to Bounds on the Secret Key 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.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.

arXiv:1706.04295EntropyOct 14, 20175 citations

On the Information Rate of Secret Sharing Schemes (Extended Abstract)

Carlo Blundo, Alfredo De Santis, Luisa Gargano +1 more

International Cryptology ConferenceAug 16, 199225 citations

Common Information and Secret Key Capacity

Himanshu Tyagi

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 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.

arXiv:1304.2444IEEE Transactions on Information TheoryMay 20, 201370 citations

Bounds on the Information Ratios of Secret Sharing Schemes for Close Access Structures.

Oriol Farràs, Jordi Ribes-González, Sara Ricci

Jan 1, 20160 citations

Secret Key Capacity under Maximal Leakage

Yingxin Zhang, Lin Zhou

IEEE International Symposium on Information Theory (ISIT)Jun 1, 20260 citations

Secret Key Agreement via Secure Omniscience

Praneeth Kumar Vippathalla, Chung Chan, Navin Kashyap +1 more

In this paper, we explore the connection between secret key agreement and secure omniscience within the setting of the multiterminal source model with an eavesdropper having side information. While the secret key agreement problem considers the generation of a maximum-rate secret key through public discussion, the secure omniscience problem is concerned with communication protocols for omniscience that minimize the rate of information leakage to the eavesdropper. The starting point of our work is a lower bound on the minimum leakage rate for omniscience, $R_{ \text {L}}$ , in terms of the wiretap secret key capacity, $C_{ \text {W}}$ . Our interest is in identifying broad classes of sources for which this lower bound is met with equality, in which case we say that there is a duality between secure omniscience and secret key agreement. We show that this duality holds in the case of certain finite linear source (FLS) models, such as two-terminal FLS models and pairwise independent network models on trees with a linear eavesdropper. Duality also holds for any FLS model in which $C_{ \text {W}}$ is achieved by a perfect linear secret key agreement scheme. We conjecture that the duality in fact holds unconditionally for any FLS model. On the negative side, we give an example of a (non-FLS) source model for which duality does not hold if we limit ourselves to communication-for-omniscience protocols with at most two (interactive) communications. We also address the secure function computation problem and explore the connection between the minimum leakage rate for computing a function and the wiretap secret key capacity.In this paper, we explore the connection between secret key agreement and secure omniscience within the setting of the multiterminal source model with an eavesdropper having side information. While the secret key agreement problem considers the generation of a maximum-rate secret key through public discussion, the secure omniscience problem is concerned with communication protocols for omniscience that minimize the rate of information leakage to the eavesdropper. The starting point of our work is a lower bound on the minimum leakage rate for omniscience,$R_{ \text {L}}$, in terms of the wiretap secret key capacity,$C_{ \text {W}}$. Our interest is in identifying broad classes of sources for which this lower bound is met with equality, in which case we say that there is a duality between secure omniscience and secret key agreement. We show that this duality holds in the case of certain finite linear source (FLS) models, such as two-terminal FLS models and pairwise independent network models on trees with a linear eavesdropper. Duality also holds for any FLS model in which$C_{ \text {W}}$is achieved by a perfect linear secret key agreement scheme. We conjecture that the duality in fact holds unconditionally for any FLS model. On the negative side, we give an example of a (non-FLS) source model for which duality does not hold if we limit ourselves to communication-for-omniscience protocols with at most two (interactive) communications. We also address the secure function computation problem and explore the connection between the minimum leakage rate for computing a function and the wiretap secret key capacity.

IEEE Transactions on Information TheoryOct 24, 20230 citations

R_{ \text {L}}

, in terms of the wiretap secret key capacity,

C_{ \text {W}}

. Our interest is in identifying broad classes of sources for which this lower bound is met with equality, in which case we say that there is a duality between secure omniscience and secret key agreement. We show that this duality holds in the case of certain finite linear source (FLS) models, such as two-terminal FLS models and pairwise independent network models on trees with a linear eavesdropper. Duality also holds for any FLS model in which

C_{ \text {W}}

is achieved by a perfect linear secret key agreement scheme. We conjecture that the duality in fact holds unconditionally for any FLS model. On the negative side, we give an example of a (non-FLS) source model for which duality does not hold if we limit ourselves to communication-for-omniscience protocols with at most two (interactive) communications. We also address the secure function computation problem and explore the connection between the minimum leakage rate for computing a function and the wiretap secret key capacity.