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Non-injective functions and one-way functions aren't the same thing. In laymen's terms, a one way function is one for which finding any element of the preimage of a given image element is difficult. (The word "difficult" is a technical term here). So, for example, SHA-256 seems to be a one-way function... Because if I give you give you a random 256 bits, it is difficult (in the technical sense of the word) for you to find ANY input to SHA-256 that will output the random numbers I gave you. We have several candidates for one-way functions that seem to be one-way. But nobody has been able to present a formal proof. Interesting things: the existence of one-way functions would imply that P does not equal NP. a great deal of modern provable cryptography rests on the underlying assumption that one-way functions exist. f(x)=0 is not one-way because it is trivially easy to find a preimage element for the only element of the image (0). So a function being non-injective, or "lossy" isn't enough.

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