# Universal family of hash functions

How to prove that a $k$-universal family of hash functions is $(k-1)$-universal family?

I tried to prove it by definition of k-universal family of hash functions but I didn't know how to use the definition.

If I prove it, Is it necessary that a $1$-universal family of hash function is universal family?

Definitions:

Let $U$ be a universe of keys, and let $H$ be a finite collection of hash functions mapping $U$ to $\{0,\dots,m-1\}$.

$H$ is universal if $\forall x,y\in U$ where $x\neq y$: $\Pr[h(x)=h(y)] \leq \frac{1}{m}$ where $h$ is chosen randomly from $H$.

$H$ is k-universal if $\forall x_1,x_2,\dots,x_k\in U$ distinct elements and $\forall i_1,i_2,\dots,i_k\in \{0,\dots,m-1\}$: $\Pr[h(x_1)=i_1 \land h(x_2)=i_2 \land \dots \land h(x_k)=i_k] = \frac{1}{m^k}$ where $h$ is chosen randomly from $H$.

## 1 Answer

For your first question, here is a hint: $$\Pr[h(x_1) = i_1 \land \cdots \land h(x_{k-1}) = i_{k-1}] = \\ \sum_{i_k} \Pr[h(x_1) = i_1 \land \cdots \land h(x_{k-1}) = i_{k-1} \land h(x_k) = i_k].$$

Regarding the connection between universal and $k$-universal: Consider the family consisting of all constant functions. This family is 1-universal but not universal (for $|U|>1$).

However, it turns out that every 2-universal family is universal (exercise). The converse isn't true: for example, the family of all functions $h$ such that $h(x_1) = 0$ is universal but not 2-universal (for $m>1$).