Summary form only given. Any implementation of Carter-Wegman universal hashing from n-b strings to m-b 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(log n) parallel time bound on a CREW PRAM; and exponential size for constant depth circuits. The results on VLSI implementation are proved using information transfer bounds derived from the definition of a universal family of hash functions.