On the program size of perfect and universal hash functions

We address the question of program size of of perfect and universal hash functions. We prove matching upper and lower bounds (up to constant factors) on program size. Furthermore, we show that minimum or nearly minimum size programs can be found efficiently. In addition, these (near) minimum size programs have time complexity at most O(log* N) where N is the size of the universe in the case of perfect hashing, and time complexity 0(1) in the case of universal hashing. Thus for universal hashing programs of minimal size and minimal time complexity have been found.