Does there exist some partial" universal hashing?

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?

• 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.
– D.W.
Commented Jul 16, 2023 at 7:16

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 $$X \to Y$$.
• 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 $Y$? Commented Jul 16, 2023 at 6:35