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Wikipedia says something surprising: if we are using quadratic hashing, we can alternate the sign of the offset to avoid collisions. I can't find any counterexamples, but I can't prove it, either.

I've simplified the question to this:

Given that $M$ is a prime number, and $M \equiv 3\ (\textrm{mod}\ 4)$,

Can you prove that the function $h(k) = (-1)^k k^2 (\textrm{mod}\ M)$ is injective over the domain of integers from $0$ to $M - 1$? (Can you prove that every input maps to a distinct output?)

See also: https://stackoverflow.com/questions/49074806/preventing-secondary-clustering-in-quadratic-probing-proof

Nick ODell
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    Sketch: note that that $i, M-i$ have the same square but, since $M$ is odd they have different parity. Thus $h(i)\neq h(M-i)$. Thus there can't be solutions to $h(k)=h(l)$ of the same parity. As for opposite parity, that would imply $-k^2 \equiv l^2$ which would give us a square root of $-1$ which doesn't exist for $M\equiv 3 \pmod 4$. – lulu Jun 20 '18 at 00:56
  • https://en.wikipedia.org/wiki/Quadratic_residue#Prime_modulus See the last line – soktinpk Jun 20 '18 at 00:59
  • @lulu I think there's an error in this part: Thus $h(i)\neq h(M−i)$. Counterexample: $M = 9, h(3) = 0, h(6) = 0$. (M is not prime and does not have residue here, but the first part of your proof does not depend on that.) – Nick ODell Jun 20 '18 at 01:16
  • @NickODell That counterexample only makes sense for composite $M$. For prime $M$, it is impossible to have a non-trivial square root of $0$. – lulu Jun 20 '18 at 09:57

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This just expands on lulu's answer. (Note: if the $\text{mod}$ is left out, assume $\pmod M$).

First, notice that for a prime modulus $M$, every residue has exactly two roots. Every residue is congruent to $r^2$ for some $r$. To prove that the only solutions to $x^2\equiv r^2$ are $r$ and $-r$ and there is no third solution, we can factor to $(x-r)(x+r)\equiv 0$. If $x$ is not equal to $r$ or $-r$, then the left hand side is the product of two nonzero numbers, which is nonzero modulo a prime number (this is not true if $M$ is not prime, e.g. $3\cdot 3\equiv 0\pmod 9$, which leads to the counterexample in the comments).

Thus, if $l$ and $k$ are collisions of the same parity, we would have $l^2\equiv k^2$, which would imply $l=M-k$, but that is impossible as they are of the same parity and $M$ is odd.

Similarly, if they were collisions of opposite parity, then $-k^2\equiv l^2$. Then multiplying by $\left(k^{-1}\right)^2$ on both sides, $-1\equiv \left(l\cdot k^{-1}\right)^2$, which is impossible when $M\equiv 3\pmod 4$.

soktinpk
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