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2

Someone can find a preimage (or prove that there is no such preimage) with about $2^{20}$ trial squares, and no precomputed storage. ACtually, I believe that the below procedure will actually achieve $2^{18}$ trial squares; that requires closer analysis than I feel like at the moment. Here is the key observation that we can take advantage of to show this: ...

4

This is probably not secure enough for a proof of work. I'll outline some attack, of increasing sophistication/complexity and increasing effectiveness (decreasing runtime). Brute force The obvious attack is brute force: enumerate all $2^{32}$ possible inputs and check to find the first that produces the desired output. This takes $2^{32}$ time. I'm sure ...

2

Say $m$ is the number and $h=f(m)$ it will be pretty easy to find $m'$ (not necessarily equal to $m$) such that $f(m)=f(m')$ on a modern computer. Brute Force The output of $f(m)$ is 32 bits. The following python function will do it def find_collision(val): while True: test = random.getrandbits(32) target = ((test*test) >> 16) & 0xffffffff) ...

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