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I was wondering, whether there are any problems that we already know are difficult to solve for a quantum computer, and that we could potentially use in cryptography, just as we do now with e.g. the factorization of integers?

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As far as I know a cipher with random key cannot be broken, however, in classical realm it is problematic to distribute the key without being eavesdropped. Probably physical exchange of the key would be necessary which impractical for digital communication.

As to break random key cipher is not possible according to theory, it is not possible to do so even on a quantum computer. So there is a task difficult (or rather impossible) for a quantum computer which can be used in cryptography.

But at the same time, quantum computers are used to implement the cipher. A quantum realm allows to create and distribute the random key and the same time being able to realize whether you were eavesdropped or not. See BB84 protocol for more information how this is done.

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