This paper studies an information-theoretic one-shot 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,Yn) are observed sequentially and the key is output bit by bit, or in situations where the random source is not an i.i.d. or ergodic process.
This paper studies an information-theoretic one-shot 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,Yn) are observed sequentially and the key is output bit by bit, or in situations where the random source is not an i.i.d. or ergodic process.
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.
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.
Abstract. We consider secret key agreement by multiple parties observ-ing correlated data and communicating interactively over an insecure communication channel. Our main contribution is a single-shot upper bound on the length of the secret keys that can be generated, with-out making any assumptions on the distribution of the underlying data. Heuristically, we bound the secret key length in terms of “how far ” is the joint distribution of the initial observations of the parties and the eaves-dropper from a distribution that renders the observations of the parties conditionally independent across some partition, when conditioned on the eavesdropper’s side information. The closeness of the two distribu-tions is measured in terms of the exponent of the probability of error of type II for a binary hypothesis testing problem, thus bringing out a structural connection between secret key agreement and binary hypoth-esis testing. When the underlying data consists of an independent and identically distributed sequence, an application of our bound recovers
We consider a non-asymptotic (one-shot) version of the multiterminal secret key agreement problem on a finite linear source model. In this model, the observation of each terminal is a linear function of an underlying random vector composed of finitely many i.i.d. uniform random variables. By restricting the public discussion to be a linear function of the terminals' observations, we obtain a characterization of the communication complexity (minimum number of symbols of public discussion) of generating a secret key of maximum length. More precisely, we show that the minimum discussion can be achieved by a non-interactive protocol in which each terminal first does a linear processing of its own private observations, following which the terminals all execute a discussion-optimal communication-for-omniscience protocol. The secret key can be chosen to be a linear function of the vector of all observations.
The multiterminal secret key agreement problem by public discussion is formulated with an additional source compression step where, prior to the public discussion phase, users independently compress their private sources to filter out strongly correlated components in order to generate a common secret key. The objective is to maximize the achievable key rate as a function of the joint entropy of the compressed sources. Since the maximum achievable key rate captures the total amount of information mutual to the compressed sources, an optimal compression scheme essentially maximizes the multivariate mutual information per bit of randomness of the private sources, and can therefore be viewed more generally as a dimension reduction technique. Single-letter lower and upper bounds on the maximum achievable key rate are derived for the general source model, and an explicit polynomial-time computable formula is obtained for the pairwise independent network model. In particular, the converse results and the upper bounds are obtained from those of the related secret key agreement problem with rate-limited discussion. A precise duality is shown for the two-user case with one-way discussion, and such duality is extended to obtain the desired converse results in the multi-user case. In addition to posing new challenges in information processing and dimension reduction, the compressed secret key agreement problem helps shed new light on resolving the difficult problem of secret key agreement with rate-limited discussion by offering a more structured achieving scheme and some simpler conjectures to prove.