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IEEE International Symposium on Information Theory (ISIT)

A new measure for conditional mutual information and its properties

Renato Renner, J. Skripsky, Stefan Wolf

Published Jan 1, 2003• 6 citations
Publisher

Abstract

We propose a new conditional mutual information measure, called the reduced intrinsic in- formation, and show its significance in the context of determining the number of secret-key bits that can be extracted from distributed information by public communication. I. THE REDUCED INTRINSIC INFORMATION The secret-key rate S(X; YllZ) of a tripartite probability distribution PXYZ is the rate at which two parties, knowing realizations of X and Y, respectively, can generate, by pub- lic communication, common bits about which a third party, who has access to 2, remains almost completely ignorant (l). It is a fundamental problem to express S(X;YllZ) in terms of Pxyz. In (2), the intrinsic information I(X;YJ. Z) := infpZlz(I(X;YIZ)) was shown to be an upper bound on S(X; YllZ). (Here, the infimum is taken over all possible ways the third party Eve can process her information 2.)

DOI
10.1109/isit.2003.1228273

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We propose a new conditional mutual information measure, called the reduced intrinsic in- formation, and show its significance in the context of determining the number of secret-key bits that can be extracted from distributed information by public communication. I. THE REDUCED INTRINSIC INFORMATION The secret-key rate S(X; YllZ) of a tripartite probability distribution PXYZ is the rate at which two parties, knowing realizations of X and Y, respectively, can generate, by pub- lic communication, common bits about which a third party, who has access to 2, remains almost completely ignorant (l). It is a fundamental problem to express S(X;YllZ) in terms of Pxyz. In (2), the intrinsic information I(X;YJ. Z) := infpZlz(I(X;YIZ)) was shown to be an upper bound on S(X; YllZ). (Here, the infimum is taken over all possible ways the third party Eve can process her information 2.)
This paper is concerned with secret-key agreement by public discussion: two parties Alice and Bob and an adversary Eve have access to independent realizations of random variables X , Y , and Z, respectively, with joint distribution PXY Z . The secret key rate S(X ; Y jjZ) has been defined as the maximal rate at which Alice and Bob can generate a secret key by communication over an insecure, but authenticated channel such that Eve's information about this key is arbitrarily small. We define a new conditional mutual information measure, the intrinsic conditional mutual information between X and Y when given Z, denoted by I(X ; Y#Z), which is an upper bound on S(X ; Y jjZ). The special scenarios where X , Y , and Z are generated by sending a binary random variable R, for example a signal broadcast by a satellite, over independent channels, or where Z is generated by sending X and Y over erasure channels, are analyzed. In the first scenario it can be shown that the secret key rate is str...
This paper is concerned with secret-key agreement by public discussion. Assume that two parties Alice and Bob and an adversary Eve have access to independent realizations of random variables X, Y, and Z, respectively, with joint distribution P/sub XYZ/. The secret-key rate S(X;Y/spl par/Z) has been defined as the maximal rate at which Alice and Bob can generate a secret key by communication over an insecure, but authenticated channel such that Eve's information about this key is arbitrarily small. We define a new conditional mutual information measure, the intrinsic conditional mutual information between S and Y when given Z, denoted by I(X;Y/spl darr/Z), which is an upper bound on S(X;Y/spl par/Z). The special scenarios are analyzed where X, Y, and Z are generated by sending a binary random variable R, for example a signal broadcast by a satellite, over independent channels, or two scenarios in which Z is generated by sending X and Y over erasure channels. In the first two scenarios it can be shown that the secret-key rate is strictly positive if and only if I(X;Y/spl darr/Z) is strictly positive. For the third scenario, a new protocol is presented which allows secret-key agreement even when all the previously known protocols fail.
The secret-key rate measures the rate at which Alice and Bob can extract secret bits from sampling a joint probability distribution, unknown to an eavesdropper Eve. The secret-key rate has been bounded above by the intrinsic information and reduced intrinsic information. However, we prove that the reduced intrinsic information is 0 if and only if the intrinsic information is 0. This result implies that at least one of the following two conjectures is false: bound secrecy exists, or the reduced intrinsic information equals the secret-key rate. We give an explicit construction of an information-erasing binarization for a candidate for bound secrecy. We then introduce some approaches for proving the existence of bound secrecy, such as reducing the channel space, linearly transforming Bob's map, and perturbing a channel for Eve.
One big problem of the communication between two parties is the secrecy. That means how\nmuch information a third party can obtain by intercepting the messages transmitted from\none honest party to the other one. Therefore cryptography offers a wide range of protocols\nto ensure security with assumptions on the eavesdropper. So one was looking for an\ninformation-theoretical description of the scenario to get unconditional secure communication.\nIn this scenario we are considering two honest parties that want to communicate over\nan authenticated channel that the eavesdropper is wiretapping. This scenario introduced the definition of the intrinsic information and the secret-key rate which are a measure of the secrecy in this setting. Later because of strong analogies to quantum mechanics it turned out that this description was lacking a phenomena called bound information which is the disability of a probability distribution to create a secret-key even though it has predicted secrecy.\nNearly ten years of research have shown the existence of bound information for the multipartite case where several parties are communicating but not yet for the bipartite case.\nHence the approach of non-distillability seems a very promising one to find this conjecture.\nMotivated by this the approach we implemented this tool and simulated some distributions that have conjectured bound information. Thereby we improved the tool to reduce its calculationtime and to get closer to the aim.
This is the first part of a three-part paper on secret-key agreement secure against active adversaries. In all three parts, we address the question whether two parties, knowing some correlated pieces of information X and Y, respectively, can generate a string S about which an adversary, knowing some information Z and having read and write access to the communication channel used by the legitimate partners, is almost completely ignorant. Whether such key agreement is possible, and if yes at which rate, is an inherent property of the joint probability distribution P/sub XYZ/. In this part, we first prove a number of general impossibility results. We then consider the important special case where the legitimate partners as well as the adversary have access to the outcomes of many independent repetitions of a fixed tripartite random experiment. In this case, the result characterizing the possibility of secret-key agreement secure against active adversaries is of all-or-nothing nature: either a secret key can be generated at the same rate as in the (well-studied) passive-adversary case, or such secret-key agreement is completely impossible. The exact condition characterizing the two cases is presented.