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. 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...
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.>
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.>
We study the problem of information-theoretically secure secret key agreement under the well-known source model and channel model. In both of these models the parties wish to create a shared secret key that is secure from an eavesdropper with unlimited computational resources. In the channel model, the first party can choose a sequence of inputs to a discrete memoryless channel, which has outputs at the other parties and at the eavesdropper. After each channel use, the parties can engage in arbitrarily many rounds of interactive authenticated communication over a public channel. At the end, each party should be able to generate the key. In the source model, the parties wishing to generate a secret key (as well as the eavesdropper) receive a certain number of independent identically distributed copies of jointly distributed random variables after which the parties are allowed interactive authenticated public communication, at the end of which each party should be able to generate the key. We derive new lower and upper bounds on the secret key rate under the source model and the channel model, and introduce a technique for proving that a given expression bounds the secrecy rate from above in the channel model. Our lower bounds strictly improve what is essentially the best known lower bound in both the source model and the channel model. Our upper bound in the channel model strictly improves the current state of art upper bound. We do not know whether our new upper bound in the source model represents an strict improvement but it includes the current best known bound as a special case.
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 studies an information-theoretic oneshot variable-length secret key agreement problem with public discussion. Let X and Y be jointly distributed random variables, each taking values in some measurable space. Alice and Bob observe X and Y respectively, can communicate interactively through a public noiseless channel, and want to agree on a key length and a key that is approximately uniformly distributed over all bit sequences with the agreed key length. The public discussion is observed by an eavesdropper, Eve. The key should be approximately independent of the public discussion, conditional on the key length. We show that the optimal expected key length is close to the mutual information I(X; Y ) within a logarithmic gap. Moreover, an upper bound and a lower bound on the optimal expected key length can be written down in terms of I(X; Y ) only. This means that the optimal one-shot performance is always within a small gap of the optimal asymptotic performance regardless of the distribution of the pair (X, Y ). This one-shot result may find applications in situations where the components of an i.i.d. pair source (Xn, Y n) are observed sequentially and the key is output bit by bit with small delay, or in situations.