Universal Hash Functions

We want to select from a set of hash functions H with reasonable certainty that the above property is true.

Thus, the number of functions $\mid$f$\mid$ in H such that h(x) = h(y) for x,y $\in$ U must satisfy

$$\frac{\mid f \mid}{\mid H \mid} \;=\; \frac{1}{m} \;\longrightarrow\; \mid f \mid \;=\; \frac{\mid H \mid}{m}$$

Definition: A universal collection of hash functions H contains exactly |H|/m hash functions such that h(x) = h(y) for x,y $\in$ U.

$$h_a(x) \;=\; \sum_{i=0}^{r} a_i x_i \; mod \; m$$

where key x = <x₀, x₁, .., x_r> is decomposed into r+1 bytes
a = <a₀, a₁, .., a_r>, each chosen randomly from {0, 1, .., m-1}.

is a universal collection of hash functions.

Thus, we want to randomly select ``a'' each time.