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Many definitions related to universal hashing mention the term

a family of hash functions, say $H$

What does this 'family' mean exactly? A numerical example would be appreciated. Also, what does it mean to choose a hash function h at random from a hash family $H$?

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The 'family' means that there is actually an extra input, which is usually regarded as fixed.

For any finite field $F, H : F\times F^{2} \to F$ given by $H\left(k,\langle x,y\rangle\right) = (k\cdot x)+y$ is a universal hash family.

It means to choose the first input at random.

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In general, having a family of functions means that you have a function definition with a small detail, change, parameter or other tweak that can vary from function to function.

An (insecure) example would be if you'd parametrize the S-box in AES. Then you have a family of functions, and you select a particular member of that family by specifying the S-box. If you'd specify the S-box to be equal to the AES S-box you'd have AES.

Selecting a member of a function family at random means that you select the parameter at random. An example would be to randomly generate the S-box byte by byte.

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  • $\begingroup$ For a block cipher, the usual "family parameter" is the key (this is why a block cipher is also a pseudo-random permutation family). Do you have an example for the parameter with actual hash functions? $\endgroup$
    – Paŭlo Ebermann
    Commented Sep 11, 2013 at 19:24
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    $\begingroup$ @PaŭloEbermann BLAKE2 has a 8/16 byte personalization string, which you can see as a parameter selecting a hash function out of the family of BLAKE2 hash functions. But that's not it's only parameter - it also takes a salt and key. $\endgroup$
    – orlp
    Commented Sep 11, 2013 at 19:29

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