# An easy computable injection with a hard inverse

I was reading Charles Bennett's Thermodynamics of Computer Science and a passage (p. 926) caught my eye

The construction of a reversible machine from an irreversible machine implies that the open question, of whether there exists a 1:1 function much easier to compute by an irreversible machine than by any reversible machine, is equivalent to the question of whether there is an easy 1:1 function with a hard inverse.

I was quite surprised the existence of an easy 1:1 function with a hard inverse was open, and even more surprised at how hard it seemed to find sources on this (though I'm very unfamiliar with this area, so it's wholly possible I was just searching the wrong things). Seeing as this paper is almost 40 years old, I was wondering if this question was still open, and whether any sources discuss it in more depth.

• Is this different from the question of whether or not one-way permutations exist? Oct 26, 2021 at 23:08
• @YonatanN This seems to be a weaker definition, as one-way functions allow for negligible error. So I suppose there's no way the problem has been answered positively, but it doesn't seem impossible that it's been answered negatively. Thank you for the reference though, it's much closer to anything I found, surprised I missed it! Oct 26, 2021 at 23:14
• Oh, it seems this is exactly the definition of worst-case one-way function, rather than the more usual average-case one-way function, the existence of which is known to be equivalent to P=NP, so I suppose it is open :) Oct 26, 2021 at 23:20

Following some direction provided by Yonatan's comment it seems this is exactly the problem of the existence of a worst-case one-way function, which is equivalent to $$P\ne NP$$
• $P \neq NP$ is not known to imply the existence of one-way functions. This is a long-standing open problem in computational complexity. Oct 27, 2021 at 11:47
• @MartinBerger I know $P\ne NP$ does not imply the existence of the much more common average-case one-way functions, but sources I have found state that it does imply the existence of worst-case one-way functions (see the reference given). Is this wrong? Oct 28, 2021 at 1:46
• Interesting, that still seems like the wrong notion. This answer lead me to believe that worst-case OWFs were the deterministic equivalent of average-case ones, and the paper by Selman was linked elsewhere and seems to use the definition I was looking for. I'm asking about a deterministic polytime function with no deterministic polytime inverse (up to trivialities involving large inputs with small outputs), which seems to be what Selman discusses and claims exist iff $P\ne NP$, though it seems these may not be called worst case OWFs. Oct 28, 2021 at 17:27
• @JoshuaGrochow makes a slightly different claim: worst-case one-way functions exist if and only if $𝖯 \neq 𝖴𝖯$ with the relationship between $NP$ and $UP$ being open. Oct 28, 2021 at 21:35