We consider two hashing models for storing a set S ⊂ {0, 1, 2, …, m - 1} in a table T of size n.The first model uses universal hashing for a partially loaded table. A set of hash functions is universal if, for any the input set, a randomly selected function has an efficient expected performance. Universal hash functions originate in [CW79], where they were used for open hashing using chaining. [CW79] poses as an open question whether comparable results can be achieved for any closed hashing schemes.The second model is perfect hashing for a full table. In preprocessing the input set is used to determine a hash function that achieves some desired performance criteria. This model was used among others in [ME82] and [FKS84].In both models a key problem is to construct a “small” set of functions, which will permit a short description (program) for each function in the set.We show, for the first time, that universal hashing can be successfully used for closed hashing and in particular for double hashing. Specifically, the set of congruential polynomials of O(log n) degree is universal for double hashing if the table load is below .75; the program size (or number of random bits generated by the algorithm) is O(log log m + log2n).For perfect hashing, we obtain nearly tight results on the size of oblivious O(1)-probe hash functions: Oblivious k-probe hash functions require O(n/k2e-k + log log m) bits of description.A probabilistic construction is presented, which shows that oblivious k-probe hash functions, can be specified in O(ne-k + log log m) bits, which nearly matches the above lower bound.We give a variation of an O(1) time 1-probe (perfect) hash function that can be specified in O(n + log log m) bits, which is tight to within a constant factor of the lower bound.In view of the adaptive schemes presented in [FNSS88], these bounds establish a significant gap between oblivious and non-oblivious O(1)-probe search.