Any implementation of Carter-Wegman universal hashing from n-bit strings to m-bit strings requires a time-space tradeoff of TS=Ω(nm). The bound holds in the general boolean branching program model and, thus, in essentially any model of computation. As a corollary, computing a + b * c in any field F requires a quadratic time-space tradeoff, and the bound holds for any representation of the elements of the field. Other lower bounds on the complexity of any implementation of universal hashing are given as well: quadratic AT2 bound for VLSI implementation; Ω(logn) parallel time bound on a CREW PRAM; and exponential size for constant-depth circuits.
Bibliographical noteFunding Information:
* Supported by an IBM graduate fellowship. ** Research was done while the author was at Laboratory for Computer Science, MIT, 545 Tech. Sq., Cambridge, MA 02139, USA. Partially supported by NSF 865727-CCR and AR0 DALL03-86-K-017. *** This work was performed while the author was at IBM Research Division, T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA.