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Does there exist a function $f(x,y)$ with these properties:

  1. Computing $f(x,y)$ is in P.

  2. $f$ is associative: $f(x, f(y, z)) = f(f(x, y), z)$.

  3. $f$ is one-way (assuming P $\neq$ NP): Given the value $v$ of some $f(x, y)$, zero or one of the values $x$ and $y$, and given that there exists a solution, it is difficult to compute a $z$ and $w$ for which $f(z,w) = v$.

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    $\begingroup$ I find it hard to call $f$ a cryptographic hash function, because while I don't have an example that violates your one-way property, collision resistance is trivially broken: $$f(f(v, x), y) = f(x, f(y, v)).$$ $\endgroup$
    – orlp
    Mar 24 at 20:28
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    $\begingroup$ crypto.stackexchange.com/q/17935/351 $\endgroup$
    – D.W.
    Mar 25 at 23:16

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