On Minimal Tree Realizations of Linear Codes

A tree decomposition of the coordinates of a code is a mapping from the coordinate set to the set of vertices of a tree. A tree decomposition can be extended to a tree realization, i.e., a cycle-free realization of the code on the underlying tree, by specifying a state space at each edge of the tree, and a local constraint code at each vertex of the tree. The constraint complexity of a tree realization is the maximum dimension of any of its local constraint codes. A measure of the complexity of maximum-likelihood decoding for a code is its treewidth, which is the least constraint complexity of any of its tree realizations. It is known that among all tree realizations of a code that extends a given tree decomposition, there exists a unique minimal realization that minimizes the state space dimension at each vertex of the underlying tree. In this paper, we give two new constructions of these minimal realizations. As a by-product of the first construction, a generalization of the state-merging procedure for trellis realizations, we obtain the fact that the minimal tree realization also minimizes the local constraint code dimension at each vertex of the underlying tree. The second construction relies on certain code decomposition techniques that we develop. We further observe that the treewidth of a code is related to a measure of graph complexity, also called treewidth. We exploit this connection to resolve a conjecture of Forney's regarding the gap between the minimum trellis constraint complexity and the treewidth of a code. We present a family of codes for which this gap can be arbitrarily large.Comment: Submitted to IEEE Transactions on Information Theory; 29 pages, 11 figure

Constraint Complexity of Realizations of Linear Codes on Arbitrary Graphs

A graphical realization of a linear code C consists of an assignment of the coordinates of C to the vertices of a graph, along with a specification of linear state spaces and linear ``local constraint'' codes to be associated with the edges and vertices, respectively, of the graph. The \k-complexity of a graphical realization is defined to be the largest dimension of any of its local constraint codes. \k-complexity is a reasonable measure of the computational complexity of a sum-product decoding algorithm specified by a graphical realization. The main focus of this paper is on the following problem: given a linear code C and a graph G, how small can the \k-complexity of a realization of C on G be? As useful tools for attacking this problem, we introduce the Vertex-Cut Bound, and the notion of ``vc-treewidth'' for a graph, which is closely related to the well-known graph-theoretic notion of treewidth. Using these tools, we derive tight lower bounds on the \k-complexity of any realization of C on G. Our bounds enable us to conclude that good error-correcting codes can have low-complexity realizations only on graphs with large vc-treewidth. Along the way, we also prove the interesting result that the ratio of the \k-complexity of the best conventional trellis realization of a length-n code C to the \k-complexity of the best cycle-free realization of C grows at most logarithmically with codelength n. Such a logarithmic growth rate is, in fact, achievable.Comment: Submitted to IEEE Transactions on Information Theor

On the Communication Complexity of Secret Key Generation in the Multiterminal Source Model

Communication complexity refers to the minimum rate of public communication required for generating a maximal-rate secret key (SK) in the multiterminal source model of Csiszar and Narayan. Tyagi recently characterized this communication complexity for a two-terminal system. We extend the ideas in Tyagi's work to derive a lower bound on communication complexity in the general multiterminal setting. In the important special case of the complete graph pairwise independent network (PIN) model, our bound allows us to determine the exact linear communication complexity, i.e., the communication complexity when the communication and SK are restricted to be linear functions of the randomness available at the terminals.Comment: A 5-page version of this manuscript will be submitted to the 2014 IEEE International Symposium on Information Theory (ISIT 2014

The Treewidth of MDS and Reed-Muller Codes

The constraint complexity of a graphical realization of a linear code is the maximum dimension of the local constraint codes in the realization. The treewidth of a linear code is the least constraint complexity of any of its cycle-free graphical realizations. This notion provides a useful parametrization of the maximum-likelihood decoding complexity for linear codes. In this paper, we prove the surprising fact that for maximum distance separable codes and Reed-Muller codes, treewidth equals trelliswidth, which, for a code, is defined to be the least constraint complexity (or branch complexity) of any of its trellis realizations. From this, we obtain exact expressions for the treewidth of these codes, which constitute the only known explicit expressions for the treewidth of algebraic codes.Comment: This constitutes a major upgrade of previous versions; submitted to IEEE Transactions on Information Theor

A Lattice Coding Scheme for Secret Key Generation from Gaussian Markov Tree Sources

In this article, we study the problem of secret key generation in the multiterminal source model, where the terminals have access to correlated Gaussian sources. We assume that the sources form a Markov chain on a tree. We give a nested lattice-based key generation scheme whose computational complexity is polynomial in the number, N , of independent and identically distributed samples observed by each source. We also compute the achievable secret key rate and give a class of examples where our scheme is optimal in the fine quantization limit. However, we also give examples that show that our scheme is not always optimal in the limit of fine quantization.Comment: 10 pages, 3 figures. A 5-page version of this article has been submitted to the 2016 IEEE International Symposium on Information Theory (ISIT

Secure Compute-and-Forward in a Bidirectional Relay

We consider the basic bidirectional relaying problem, in which two users in a wireless network wish to exchange messages through an intermediate relay node. In the compute-and-forward strategy, the relay computes a function of the two messages using the naturally-occurring sum of symbols simultaneously transmitted by user nodes in a Gaussian multiple access (MAC) channel, and the computed function value is forwarded to the user nodes in an ensuing broadcast phase. In this paper, we study the problem under an additional security constraint, which requires that each user's message be kept secure from the relay. We consider two types of security constraints: perfect secrecy, in which the MAC channel output seen by the relay is independent of each user's message; and strong secrecy, which is a form of asymptotic independence. We propose a coding scheme based on nested lattices, the main feature of which is that given a pair of nested lattices that satisfy certain "goodness" properties, we can explicitly specify probability distributions for randomization at the encoders to achieve the desired security criteria. In particular, our coding scheme guarantees perfect or strong secrecy even in the absence of channel noise. The noise in the channel only affects reliability of computation at the relay, and for Gaussian noise, we derive achievable rates for reliable and secure computation. We also present an application of our methods to the multi-hop line network in which a source needs to transmit messages to a destination through a series of intermediate relays.Comment: v1 is a much expanded and updated version of arXiv:1204.6350; v2 is a minor revision to fix some notational issues; v3 is a much expanded and updated version of v2, and contains results on both perfect secrecy and strong secrecy; v3 is a revised manuscript submitted to the IEEE Transactions on Information Theory in April 201

On the Public Communication Needed to Achieve SK Capacity in the Multiterminal Source Model

The focus of this paper is on the public communication required for generating a maximal-rate secret key (SK) within the multiterminal source model of Csisz{\'a}r and Narayan. Building on the prior work of Tyagi for the two-terminal scenario, we derive a lower bound on the communication complexity, $R_{\text{SK}}$, defined to be the minimum rate of public communication needed to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by $R_{\text{CO}}$, is an upper bound on $R_{\text{SK}}$. For the class of pairwise independent network (PIN) models defined on uniform hypergraphs, we show that a certain "Type $\mathcal{S}$" condition, which is verifiable in polynomial time, guarantees that our lower bound on $R_{\text{SK}}$ meets the $R_{\text{CO}}$ upper bound. Thus, PIN models satisfying our condition are $R_{\text{SK}}$-maximal, meaning that the upper bound $R_{\text{SK}} \le R_{\text{CO}}$ holds with equality. This allows us to explicitly evaluate $R_{\text{SK}}$ for such PIN models. We also give several examples of PIN models that satisfy our Type $\mathcal S$ condition. Finally, we prove that for an arbitrary multiterminal source model, a stricter version of our Type $\mathcal S$ condition implies that communication from \emph{all} terminals ("omnivocality") is needed for establishing a SK of maximum rate. For three-terminal source models, the converse is also true: omnivocality is needed for generating a maximal-rate SK only if the strict Type $\mathcal S$ condition is satisfied. Counterexamples exist that show that the converse is not true in general for source models with four or more terminals.Comment: Submitted to the IEEE Transactions on Information Theory. arXiv admin note: text overlap with arXiv:1504.0062