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Definition (Carter Wegman 1979): Family \(H\) of hash functions is 2 -universal if for any \(x\neq y\in S\) :

\(Pr_{h}{}_{\in}{}_{H}[h(x)=h(y)]\leq\dfrac{1}{n}\)

\(\sum_{x}{}_{\neq}{}_{y}E[h(x)=h(y)]\leq{m\choose 2}\dfrac{1}{n}\)

where \(m\) is the number of eklements and \(n\) the number of hash functions in \(H\).

\({m\choose 2}\dfrac{1}{n}\rightarrow\dfrac{k!}{2(k-2)!}*\dfrac{1}{40k}=\dfrac{k!}{80k(k-2)!}=\dfrac{k^{2}-k}{80k}=\dfrac{k-1}{80}\)

If we analyse \((k-2)!\)