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Why can't we pick just $x + b$ for example? I know that for universal hashing the Pr[$h(x) == h(y)$] $\leq 1/|n|$

But with the same hash function, just without the $a$ we can get $h(k) = (($k$ + b$ mod $p$) mod $n$).

I've run through a million examples and I can't find a single reason why I might be wrong.

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You pick $p\ge n$ and your family consists of $h_b(x)=(x+b\bmod p)\bmod n$, where $b$ ranges over $0,\ldots, b-1$. To show that this family is universal, we need that for any $0\le xn+1$ and $n>k+1$, then $h_k(k+1)=0$ and $h_k(n-k)=0$. Also, $h_{k+1}(k+1)=1$ and $h_{k+1}(n-k)=1$. Thus for $x=k+1\ne n-k=y$, the probability of $h_b(x)=h_b(y)$ is at least $\frac 2{p}$. Unfortunately, this may well be $>\frac 1n$.

You'd be better of to consider $h_b(x)=x+b\bmod n$. But that would not really be hash-y after all

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    Yep, that works. Can you run me through why you chose $p = 2k + 1 > n+1$ and $n > k+1$ and how you got $x = k+1$ and $y = n-k$. I want to learn how you reached these numbers. Thanks – 2017-02-15