A new way of constructing (minimal) perfect hash functions is described. The<br />technique considerably reduces the overhead associated with resolving buckets in two-level hashing schemes. Evaluating a hash function requires just one multiplication and a few additions apart from primitive bit operations. The number of accesses to memory is two, one of which is to a fixed location. This improves the probe performance of previous minimal perfect hashing schemes, and is shown to be optimal. The hash function description ("program") for a set of size n occupies O(n) words, and can be constructed in expected O(n) time.