Extractable Common Randomness From Gaussian Trees: Topological and Algebraic Perspectives

In this paper, we study both topological and algebraic properties of unrooted Gaussian trees in order to characterize their security performance. Such performance is measured by the corresponding potential in extracting common randomness from a given tree, which is further determined by max-min and min-max conditional mutual information (CMI) values, subject to the order of selecting variables from the tree by legitimate nodes Alice and Bob, and an eavesdropper Eve, respectively. A new operation is proposed to transform a Gaussian tree into another, and also to order different Gaussian trees. Through such operation we construct several equivalent classes of Gaussian trees. Each class includes multiple Gaussian trees that can be partially ordered based on the associated max-min or min-max CMI metric, and thus, we can find the most secure and the least secure trees in each partially ordered set (poset). The union of all posets generates all possible non-isomorphic trees of the given number of variables. Then, we assign a particular polynomial to each Gaussian tree, and show that such polynomial can determine the relative security performance of the Gaussian tree with respect to other trees within the same class. In the end, based on a generalized integer partition method, we propose a novel approach to efficiently enumerate the most secure structures of all posets.