In view of their applicability to parallel and distributed computer systems, interconnection networks have been studied intensively by mathematicians, computer scientists, and computer designers. In this paper, we propose an asymptotically optimal method for connecting a set of nodes into a perfect difference network (PDN) with diameter 2, so that any node is reachable from any other node in one or two hops. The PDN interconnection scheme, which is based on the mathematical notion of perfect difference sets, is optimal in the sense that it can accommodate an asymptotically maximal number of nodes with smallest possible node degree under the constraint of the network diameter being 2. We present the network architecture in its basic and bipartite forms and show how the related multidimensional PDNs can be derived. We derive the exact average internode distance and tight upper and lower bounds for the bisection width of a PDN. We conclude that PDNs and their derivatives constitute worthy additions to the repertoire of network designers and may offer additional design points that can be exploited by current and emerging technologies, including wireless and optical interconnects. Performance, algorithmic, and robustness attributes of PDNs are analyzed in a companion paper.