P vs NP, Has anyone proved there is no such thing as a collision-less one-way function.

Question

I know that there has not been a proof against or for, the existence for a true one-way function. But i was wondering has such a thing been proven for collision-less (injective) one-way functions.

Answer 1

Comment: be careful with the terms you use. "Injective" is common, but "collision-less" is non-standard and may be confused with "collision-free," which is something else.

First, note that since what you ask is a subset of one-way functions, proving their existence would imply the existence of one-way functions. Thus, you already know that existence of injective one-way functions hasn't been established.

Now, their existence has not been disproven either: the existence of one-way permutations (i.e., one-way functions that are bijective, and thus a fortiori injective) is a common cryptographic assumption. To this day, it has not been proven nor disproven, and their existence is not known to be implied by that of one-way functions.