Exercise 11.3.6

$\star$ Let $U$ be the set of $n$-tuples of values drawn from $\mathbb{Z}p$, and let $B = \mathbb{Z}p$, where $p$ is prime. Define hash functions $hb : U \to B$ for $b \in \mathbb{Z}p$ on an input $n$-tuple $\langle a_0, a_1, \ldots, a_{n-1} \rangle$ from $U$ as

$$ h_b(\langle a_0, a_1, \ldots, a_{n-1} \rangle) = \Bigg(\sum_{j=0}^{n-1} a_j b^j \Bigg) \bmod p $$

and let $\mathscr{H} = \{h_b : b \in \mathbb{Z}_p\}$. Argue that $\mathscr{H}$ is $((n - 1)/p)$-universal according to the definition of $\epsilon$-universal in Exercise 11.3-5. (Hint: See Exercise 31.4-4.)