We present new design and analysis techniques for the synthesis of fast parallel multiplier circuits. In \cite{Vojin}, Oklobdzija, Villeger, and Lui suggested a new approach, the Three Dimensional Method (TDM), for Partial Product Reduction Tree (PPRT) design that produces multipliers which outperform the current best designs. The goal of TDM is to produce a minimum delay PPRT using full adders. This is done by carefully modelling the relationship of the output delays to the input delays in an adder, and then interconnecting the adders in a globally optimal way. Oklobdzija, et.~al. suggested a good heuristic for finding the optimal PPRT, but no proofs about the performance of this heuristic were given. We provide a formal characterization of optimal PPRT circuits and prove a number of properties about them. For the problem of summing a set of input bits within the minimum delay, we present an algorithm that produces a minimum delay circuit in time linear in the size of the inputs. Our techniques allow us to prove tight lower bounds on multiplier circuit delays. These results are combined to create a program which finds optimal TDM multiplier designs.