# Small space hash functions that are weakly but not strongly universal

This is a follow up to this this question about weakly universal hash functions

A family of hash functions $$H_w$$ is said to be weakly universal if for all $$x \ne y$$ :

$$P_{h \in H_w}(h(x) = h(y)) \leq 1/m$$

Here the function $$h:U \rightarrow [m]$$ is chosen uniformly from the family $$H$$ and we assume $$|U| > m$$.

A family of hash functions $$H_s$$ is said to be strongly universal if for all $$x \ne y$$ and $$k, \ell \in [m]$$:

$$P_{h \in H_s}(h(x) = k \land h(y) = \ell) = 1/m^2$$

I previously asked for an example of a hash function family which is weakly universal but not strongly universal. The very nice answer was:

Take $$U = [m+1]$$, and consider the functions $$h_i$$, for $$i \in [m]$$, given by $$h_i(x) = \begin{cases} x & \text{if } x \neq m+1, \\ i & > \text{if } x = m+1. \end{cases}$$ The same approach can be used for arbitrary $$|U|$$: fix the first $$m$$ coordinates, and make all other coordinates uniformly and independently random.

For arbitrary $$|U|$$, it seems that in order to represent a single hash function in practice you would need to store a lookup table of size $$|U|$$ to ensure that the hashed values are independent and truly random. Then for every key, you would look up its random value in the vast lookup table to compute the hash function.

Is there a hash function family where the hash functions require constant or log space that achieves the same result of being weakly but not strongly universal? In other words, are there any practical hash function families that are weakly not but strongly universal?

Let $$H$$ be a family of strongly universal hash functions from $$U$$ to $$[m]$$. Construct a new family of hash functions from $$U \cup \{x\} \to [m]$$ by extending all functions $$h \in H$$ with $$h(x) = 1$$. The family is weakly universal since $$h(u)$$ is distributed uniformly for every $$u \in U$$, but it is clearly not strongly universal.