We obtain a generic lower bound on the key expansion required for securely extending the domain of a UOWHF. Our lower bound holds over a large class of "natural" domain extending algorithms. A consequence of our result is the fact that the key length expansion in Shoup's algorithm is optimal for this class. Our second contribution is to obtain a finite binary tree algorithm to extend the domain of a UOWHF. The associated key length expansion is only a constant number of bits more than the minimum possible. Our finite binary tree algorithm is the first practical parallel algorithm to securely extend the domain of a UOWHF. Also the speed-up obtained by our algorithm is approximately proportional to the number of processors.