Multiterminal Secret Key Agreement

Abstract

We consider the secret key agreement problem under the multiterminal source model proposed by Csiszár and Narayan. A single-letter characterization of the secrecy capacity is desired but remains unknown except in the extreme case with unlimited public discussion and without wiretapper's side information. Taking the problem to the opposite extreme by requiring the public discussion rate to be zero asymptotically, we obtain the desired characterization under surprisingly general setting with wiretapper's side information, silent users, trusted and untrusted helpers. An immediate consequence of the result is that the capacity with nearly no discussion is the same as the capacity with no discussion, resolving a previous conjecture in the affirmative. The idea of the proof is to characterize the capacity in the special case with neither wiretapper's side information nor untrusted helpers using a multivariate extension of Gács-Körner common information, and then extend the result to the general setting by a change of scenario that turns untrusted helpers into trusted helpers. We further show how to evaluate the capacity explicitly for finite linear sources and discuss how the current result can be extended to improve and unify existing bounds on the capacity for strictly positive discussion rates. Index TermsMultiterminal secret key agreement; Gács-Körner common information; secrecy capacity; double Markov inequality; general source model; finite linear source.

Multivariate Mutual Information Inspired by Secret-Key Agreement

“…In particular, à ðfigÞ > 0 for all i 2 V. By the complementary slackness theorem, we have r à i ¼ HðZ i jZ Vnfig Þ for any optimal r à V to the primal LP and so IðZ V Þ ¼ HðZ V Þ À r à ðVÞ ¼ J D ðZ V Þ. 20 This implies (6.12f) because of (6.8).…”

“…Equation (6.12b) may not imply (6.12a) in the bivariate case. 20 Indeed, (6.12f) is stronger than necessary. Suppose we have a set of optimal solutions to the dual linear program and put their supports in S. Repeatedly for every overlapping S 1 ; S 2 2 S such that S 1 [ S 2 6 ¼ V, add S 1 \ S 2 to S. It can be argued using the submodularity of entropy that, for all S 2 S, r à ðSÞ ¼ HðZ S jZ VnS Þ for all optimal r à V to the primal LP.…”

“…There are more general models [2], [3], [15]- [20] involving the wiretapper's side information, continuous sources, or private channels. The secrecy capacity is largely unknown in these models.…”

Info-Clustering: A Mathematical Theory for Data Clustering

“…The idea is to compute the empirical entropies of subsets of random variables Z B after quantization, and use them in (3.3a) to estimate the MMI. The MMI of the quantized random variables is shown to approach the MMI of the continuous random variables in [91,Appendix B], and the details of the quantization can be found therein. However, computing the empirical joint distribution of a subset of random variables or estimating the joint entropy from the data samples takes exponential time with respect to the size of the subset [98].…”

Coding for Positive Rate in the Source Model Key Agreement Problem

“…The supremum of all achievable key rates is called the source model secretkey (SK) capacity and denoted by S(X; Y Z). Extensions to multiple parties and continuous random variables can be found in [4][5][6][7][8][9].…”