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Let H be a class of all functions, mapping M possible keys to N integers. Is it true that H is a 2-universal family? Is it a good idea to use H in applications?

I don't even know where and how to start. Can you give me a hint on how to prove this? Is it even possible to prove this?

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Can you give me a hint on how to prove this?

Let $H$ be a class of all functions which map universe $Z_n$ to $Z_m$. We will assume that, any particular mapping is done by exactly one of the functions in the class.

Total functions in $H$ = number of all possible mappings = $m^n$

Given any distinct keys $x$ and $y$, we need to find number of all possible functions (mappings) in which they collide. The remaining $n-2$ keys have total mappings: $m^{n-2}$. The $(x,y)$ pair can collide by mapping to any of the $m$ slots. So,

Total functions in $H$ where $(x,y)$ collide = $m^{n-2}\cdot m = m^{n-1}$

Thus probability of collision = $m^{n-1} / m^n = 1/m$.

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