Outline of the paper. We have divided the presentation of results into two parts, one for the source model and one for the channel model. The source model and channel model are two different (albeit related) problems, and this division cleanly separates the treatment of the two. Furthermore it leads to a reasonable number of pages for each part. This division however leads to duplications, and may blur the conceptual connection between the two models. In order to minimize the duplications, we have decided to include in part I, a unified introduction setting out all the common notation and fundamentals. Part II of the paper will not be standalone, and is written under the assumption that the reader has access to Part I. Furthermore, the conceptual connection between the two models is emphasized in part II, where the results and their derivations for the channel model are compared with those of the source model. The outline of this paper is as follows. In section II, we introduce the basic notation and the definitions for both the source model and channel model. Section III illustrates the technique used for proving the upper bounds at an intuitive level. Section IV contains the main results of this paper followed by section V, which gives the proofs, with some of the details relegated to the appendices. Appendix I contains an example showing that our upper bound for secret key capacity is strictly better than the currently best know upper bound from [7]. Appendix V contains a counterexample to the natural conjecture (which we believed for a long time while working on this problem) that the CFO rate is a concave function of the underlying joint probably distribution.
Outline of the paper. The organization of this paper is as follows. In Section II, the system model and the corresponding assumptions are described and the secret-key capacity expression is derived. Section III contains the solution to the transmitter optimization problem. In Section IV, we develop a necessary and sufficient condition under which beamforming is optimal. In Section V, we study the impact of correlation matrices on the secret-key capacity. Section VI analyzes the system in the high power regime. In Section VII, we study low-SNR behavior of the secret-key capacity. Conclusions are provided in Section VIII. Appendices contain detailed proofs. × × Notations: Throughout this paper, the symbol † indicates the conjugate transpose operator. In denotes the n n identity matrix and diag [λ1, . . . , λn] denotes the n n diagonal matrix whose entries are λ1, . . . , λn. Vectors are denoted with ≤ · | · | small bold letters while matrices are denoted with capital bold letters. The capital letter E denotes the expectation of a random quantity. We use the symbol I(·; ·) to denote the mutual information between two random variables. Hi• and H•j represent, respectively, the ith row and the jth column of a matrix H. The symbol is used to denote the determinant of a matrix and Tr( ) denotes the trace operator. We use the notation A 0 to denote that A is a positive semi-definite matrix. A 2 denotes a square-root of a matrix. We use the notation x > y to denote that the vector x majorizes the ∼ CN{ } vector y, in the sense of [21]. The notation x 0, Σ denotes that the random vector x is complex Gaussian with [24]. Therefore, in the presence of spatial correlation at all terminals, the channel matrices are modeled as [25] zero mean and covariance matrix Σ. 1/2
Outline of the paper. In Section II, criteria for a social contract for cyberspace are outlined, in Section III, fourteen serious and on-going ethical or security issues in the Internet and cyberspace are reviewed, in Section IV, the questions of whether a social contract is really necessary, whether it is possible, and also whether in a sense it already exists are explored, in Section V a draft social contract is outlined, including a review of how this draft social contract succeeds in addressing the issues presented in Section III, and, finally, in Section VI conclusions are presented.
