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Suppose we have sets $X$ and $Y$, $|X|=m$, $|Y|=n$. $H$ is a universal family of hash functions from $X$ to $Y$. Let $S\subsetneq X$ be a proper subset of $X$. Does there exist some "partial" universal hash family that is universal on $S$ to $Y$, but not necessarily universal on $X$ to $Y$? The circuit complexity of that "partial" universal hashing should be preferably lower then $H$.

Update: Does there exist construction such that taking the union of two "partial" universal hash families from S1 and S2 to Y we can get a universal hash family from S1∪S2 to Y?

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  • $\begingroup$ Please ask only one question per post. Please don't change the question after it already has been answered in a way that invalidates existing answers. Please don't use "Update:"; revise the question so it reads well for someone who encounters it for the first time. $\endgroup$
    – D.W. ♦
    Jul 16 at 7:16

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Sure, of course. Let $S$ be a single point, so $|S|=1$; then any family is trivially a universal hash on $S \to Y$, and is faster than $H$, even though it is not universal on $X \to Y$.

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  • $\begingroup$ Thank you. Does there exist construction such that taking the union of two "partial" universal hash families from $S_1$ and $S_2$ to $Y$ we can get a universal hash family from $S_1\cup S_2$ to $Y$? $\endgroup$
    – Kagura Hitoha
    Jul 16 at 6:35
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    $\begingroup$ @KaguraHitoha, this site is not designed for asking follow-up questions or new questions in the comments. If you have a new question, you can ask it with the 'Ask Question' button, but make sure to try to resolve it on your own first, provide full context, and show what progress you've made. If you want an interactive experience, you are probably better off looking for some other resource (a private tutor?), as this site is not designed for interactive responses. $\endgroup$
    – D.W. ♦
    Jul 16 at 7:16
  • $\begingroup$ OK I understand. Thank you again for your answer. $\endgroup$
    – Kagura Hitoha
    Jul 16 at 7:49

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