Polynomial hash functions are reliable

Abstract

Polynomial hash functions are well studied and widely used in various applications. They have gained popularity because of certain performances they exhibit. It has been shown that even linear hash functions are expected to have such performances. However, quite often we would like the hash functions to be reliable, meaning that they perform well with high probability; for some certain important properties even higher degree polynomials were not known to be reliable. We show that for certain important properties linear hash functions are not reliable. We give indication that quadratic hash functions might not be reliable. On the positive side, we prove that cubic hash functions are reliable. In a more general setting, we show that higher degree of the polynomial hash functions translates into higher reliability. We also introduce a new class of hash functions, which enables to reduce the universe size in an efficient and simple manner. The reliability results and the new class of hash functions are used for some fundamental applications: improved and simplified reliable algorithms for perfect hash functions and real-time dictionaries, which use significantly less random bits, and tighter upper bound for the program size of perfect hash functions.