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arXiv:1411.0729 · IEEE Transactions on Information Theory

The Private and Public Correlation Cost of Three Random Variables With Collaboration

Eric Chitambar, Min-Hsiu Hsieh, Andreas Winter

Published Mar 10, 2016• 8 citations
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Abstract

In this paper, we consider the problem of generating arbitrary three-party correlations from a combination of public and secret correlations. Two parties-called Alice and Bob-share perfectly correlated bits that are secret from a collaborating third party, Charlie. At the same time, all three parties have access to a separate source of correlated bits, and their goal is to convert these two resources into multiple copies of some given tripartite distribution P(XYZ). We obtain a single-letter characterization of the tradeoff between public and private bits that are needed to achieve this task. The rate of private bits is shown to generalize Wyner's classic notion of common information held between a pair of random variables. The problem we consider can be contrasted fruitfully with the task of secrecy formation, in which P(XYZ) is generated using public communication and local randomness but with Charlie functioning as an adversary instead of a collaborator. We describe in detail the differences between the collaborative and adversarial scenarios.

DOI
10.1109/tit.2016.2530086

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In this paper, we consider the problem of generating arbitrary three-party correlations from a combination of public and secret correlations. Two parties-called Alice and Bob-share perfectly correlated bits that are secret from a collaborating third party, Charlie. At the same time, all three parties have access to a separate source of correlated bits, and their goal is to convert these two resources into multiple copies of some given tripartite distribution P(XYZ). We obtain a single-letter characterization of the tradeoff between public and private bits that are needed to achieve this task. The rate of private bits is shown to generalize Wyner's classic notion of common information held between a pair of random variables. The problem we consider can be contrasted fruitfully with the task of secrecy formation, in which P(XYZ) is generated using public communication and local randomness but with Charlie functioning as an adversary instead of a collaborator. We describe in detail the differences between the collaborative and adversarial scenarios.
The inverse of Maurer's secret key distillation problem from (many independent realisations of) a triple of random variables X, Y, Z by two players (Alice and Bob) against an eavesdropper (Eve) is considered: the formation of the joint distribution (up to local degrading of Z) from secret key and public communication. We determine the asymptotically minimal amount of secret key for this task, and indeed the full trade-off of secret (between Alice and Bob) vs. public (shared between Alice, Bob and Eve) correlation for this problem. Our result generalises a theorem of Wyner on the "common information of a pair of random variables", which is recovered as the special case of Z being independent of XY. We investigate the secret key required as a function of the probability distribution and compare to an analogous notion based on prior shared entanglement
The problem of generating a shared secret key S by two parties knowing dependent random variables X and Y, respectively, but not sharing a secret key initially, is considered. An enemy who knows the random variable Z, jointly distributed with X and Y according to some probability distribution P/sub XYZ/, can also receive all messages exchanged by the two parties over a public channel. The goal of a protocol is that the enemy obtains at most a negligible amount of information about S. Upper bounds on H(S) as a function of P/sub XYZ/ are presented. Lower bounds on the rate H(S)/N (as N to infinity ) are derived for the case in which X=(X/sub 1/, . . ., X/sub N/), Y=(Y/sub 1/, . . ., Y/sub N/) and Z=(Z/sub 1/, . . ., Z/sub N/) result from N independent executions of a random experiment generating X/sub i/, Y/sub i/ and Z/sub i/ for i=1, . . ., N. It is shown that such a secret key agreement is possible for a scenario in which all three parties receive the output of a binary symmetric source over independent binary symmetric channels, even when the enemy's channel is superior to the other two channels.>
An efficient paradigm for multi-party computation (MPC) are protocols structured around access to shared pre-processed computational resources. In this model, certain forms of correlated randomness are distributed to the participants prior to their computation. The shared randomness is then consumed in a computation phase that involves public communication with efficient round complexity, and the computation is secure in this second phase provided the initial correlations were distributed securely. Usually the latter requires some strong setup assumptions, such as a trusted dealer and private channels. We present a novel approach for generating these correlations from entangled quantum graph states and yield information-theoretic privacy guarantees that hold against a malicious adversary, with limited assumptions. Our primary contribution is a tripartite resource state and measurement-based protocol for extracting a binary multiplication triple, a special form of shared randomness that enables the private multiplication of a bit conjunction. Here, we employ a third party as a Referee and demand only an honest pair among the three parties. The role of this Referee is weaker than that of a Dealer, as the Referee learns nothing about the underlying shared randomness that is disseminated. We prove perfect privacy for our protocol, assuming access to an ideal copy of the resource state, an assumption that is based on the existence of graph state verification protocols. Finally, we demonstrate its application as a primitive for more complex Boolean functionalities such as 1-out-of-2 oblivious transfer (OT) and MPC for an arbitrary NNN-party Boolean function, assuming access to the proper broadcasting channel.
Consider a scenario in which two parties Alice and Bob as well as an opponent Eve receive the output of a binary symmetric source (e,g. installed in a satellite) over individual, not necessarily independent binary symmetric channels. Alice and Bob share no secret key initially and can only communicate over a public channel completely accessible to Eve. The authors derive a lower bound on the rate at which Alice and Bob can generate secret-key bits about which Eve has arbitrarily little information, This lower bound is strictly positive as long as Eve's binary symmetric channel is not perfect, even if Alice's and Bob's channels are by orders of magnitude less reliable than Eve's channel.>
We study common randomness where two parties have access to i.i.d. samples from a known random source, and wish to generate a shared random key using limited (or no) communication with the largest possible probability of agreement. This problem is at the core of secret key generation in cryptography, with connections to communication under uncertainty and locality sensitive hashing. We take the approach of treating correlated sources as a critical resource, and ask whether common randomness can be generated resource-efficiently. We consider two notable sources in this setup arising from correlated bits and correlated Gaussians. We design the first explicit schemes that use only a polynomial number of samples (in the key length) so that the players can generate shared keys that agree with constant probability using optimal communication. The best previously known schemes were both non-constructive and used an exponential number of samples. In the amortized setting, we characterize the largest achievable ratio of key length to communication in terms of the external and internal information costs, two well-studied quantities in theoretical computer science. In the relaxed setting where the two parties merely wish to improve the correlation between the generated keys of length k, we show that there are no interactive protocols using o(k) bits of communication having agreement probability even as small as 2–o(k). For the related communication problem where the players wish to compute a joint function f of their inputs using i.i.d samples from a known source, we give a simultaneous message passing protocol using 2O(c) bits where c is the interactive randomized public-coin communication complexity of f. This matches the lower bound shown previously while the best previously known upper bound was doubly exponential in c. Our schemes reveal a new connection between common randomness and unbiased error-correcting codes, e.g., dual-BCH codes and their analogues in Euclidean space.