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IEEE International Symposium on Information Theory (ISIT)

The intrinsic conditional mutual information and perfect secrecy

Ueli M. Maurer, Stefan Wolf

Published Nov 22, 2002• 21 citations
Publisher

Abstract

Conditions are derived for the possibility and impossibility of information-theoretic secret-key agreement by public discussion. A new quantity, the intrinsic information, is introduced and its relationship to secret-key agreement is investigated. A new protocol is described that allows secret-key agreement in situations for which previous protocols fail.

DOI
10.1109/isit.1997.613003

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Conditions are derived for the possibility and impossibility of information-theoretic secret-key agreement by public discussion. A new quantity, the intrinsic information, is introduced and its relationship to secret-key agreement is investigated. A new protocol is described that allows secret-key agreement in situations for which previous protocols fail.
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.
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...
It is known that noise on communication channels can be a powerful resource for certain cryptographic purposes, basically by utilizing the randomness to ensure the secrecy. In this paper, we study information theoretically secure key agreement through the use of the additive white Gaussian noise channel and public discussion. We present a protocol for key agreement that defines the computation and communications between the two communicating parties to share a common secret information, about which negligible amount of information is revealed to other third parties. We discuss the signal design, secrecy and the achievable rate of the protocol.
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.)
We revisit the problem of secret key agreement using interactive public communication for two parties and propose a new secret key agreement protocol. The protocol attains the secret key capacity for general observations and attains the second-order asymptotic term in the maximum length of a secret key for independent and identically distributed observations. In contrast to the previously suggested secret key agreement protocols, the proposed protocol uses interactive communication. In fact, the standard one-way communication protocol used prior to this work fails to attain the asymptotic results above. Our converse proofs rely on a recently established upper bound for secret key lengths. Both our lower and upper bounds are derived in a single-shot setup and the asymptotic results are obtained as corollaries.