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I was recently going through the Wiki page List of unsolved Problems in Computer Science.

There was a problem which I do not understand

Do one-way functions exist ? [Is public-key cryptography possible ?]

A one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input.

A one-way function that comes to mind is a Cryptographic hash function which is used to create message digests. Where size of the message digest is relatively small as compared to the size of the message and hence it becomes very hard to reconstruct message from the message digest.

So, why does the above problem say that these functions do not exist?

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No known cryptographic hash functions are provably secure. It might seem to be hard to reconstruct an input hashing to a given hash value, but we can't prove that it is indeed very hard. We can just say that no one was able to find an algorithm accomplishing that so far, but there is no guarantee that tomorrow such an algorithm won't be found.

In cryptography there is a very exact definition of one-way functions, which implies, among else, that P$\neq$NP. While there are several specific functions which are conjectured to satisfy the definition, no one knows how to prove this. We don't even know how to prove that one-way functions exist assuming P$\neq$NP.

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