I was reading through the classic CLRS with the intention of reviewing the hash tables theory, more specifically the hash function definition I just wanted a reference to quote.

I cannot find a formal definition given but I think it's fair to say a hash function (not univerisal) $h$ is a surjective map from a set of keys $K$ to a subset of integers $U$, for each $k \in K$ we define $h(k)$ to be the hash value of $k$. From the explanation given in CLRS it seems though this restriction on $U$ (be integers) might be too restrictive, however since I think the definition has to show some practical aspects I think this might be correct.

Can you either give me: 1. A paper/book with a formal definition 2. Confirm if my definition is correct?

Thank you

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    $\begingroup$ Not every informal concept has a standard formal definition. $\endgroup$ Commented Apr 10, 2020 at 10:20
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    $\begingroup$ I'm not sure why you require your function to be surjective. $\endgroup$ Commented Apr 10, 2020 at 10:21
  • $\begingroup$ I probably don't need surjection, since you have the notion of uniform hash. So a non uniform might imply you don't hit every value in the image space. So no formal definition then? Are there examples of hash functions whose values are not integers? $\endgroup$ Commented Apr 10, 2020 at 10:29
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    $\begingroup$ It is quite common to find hash functions whose output is a bitstring. $\endgroup$ Commented Apr 10, 2020 at 10:36
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    $\begingroup$ That said, in cryptography there are formal definitions of hash functions. But that's not what you're after. $\endgroup$ Commented Apr 10, 2020 at 12:42

1 Answer 1


A hash function is used to map a set of keys to a subrange of the integers (it is used as an index into an array, in the end). So it must be (assuming zero based arrays, as in C), $h \colon \mathcal{U} \to [0, m - 1]$ if $\mathcal{U}$ is the universe of keys.

  • $\begingroup$ So the image set is actually integers by definition, right? Any reference? $\endgroup$ Commented Apr 13, 2020 at 11:56
  • $\begingroup$ @user8469759, see how it is used. $\endgroup$
    – vonbrand
    Commented Apr 13, 2020 at 17:27

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