This dissertation considers two groups of problems related to secure communication. The first line of research is devoted to theoretical problems of copyright protection of digital content. Embedding identification data in the content is a well-developed technique of content protection known under the name of fingerprinting. Schemes that provide such protection are known as fingerprinting codes in the literature. We study limits of the number of users of a fingerprinting system as well as constructions of low-complexity fingerprinting codes that support a large number of users. The second problem that is addressed in the dissertation relates to connectivity analysis of ad hoc wireless networks. One of the basic requirements in such environments is to ensure that none of the nodes are completely isolated from the network. We address the problem of characterizing threshold parameters for node isolation that enable the system designer to choose the power needed for network operation based on the outage probability of links in the network.

The methods of this research draw from coding theory, information theory and random graphs. An idea that permeates most results in this dissertation is the application of randomization both in the analysis of fingerprinting and node isolation.

The main contributions of this dissertation belong in the area of fingerprinting and are described as follows. We derive new lower and upper bounds on the optimal trade-off between the number of users and the length of the fingerprints required to ensure reliability of the system, which we call fingerprinting capacity. Information-theoretic techniques employed in our proofs of bounds on capacity originate in coding theorems for channels with multiple inputs. Constructions of fingerprinting codes draw on methods of coding theory related to list decoding and code concatenation.

We also analyze random graph models for ad hoc networks with link failures and secure sensor networks that employ randomized key distribution. We establish a precise zero-one law for node isolation in the model with link failures for nodes placed on the circle. We further generalize this result to obtain a one-law for secure sensor networks on some surfaces.