1
$\begingroup$

This is probably obvious, but I cannot find it anywhere, since all textbooks define OWFs for average-case hardness.

Do we known if worst-case one-way permutations exist assuming $\mathbf{P} \neq \mathbf{NP}$? I don't require average-case hardness, it suffices that the function is hard to invert in the worst-case. That is, assuming $\mathbf{P} \neq \mathbf{NP}$, is there a family of permutations $f_n : \{0,1\}^n \to \{ 0,1\}^n$, such that $f_n$ can be computed by circuits of size polynomial in $n$ but there is no family of poly-size circuits that inverts $f_n$ on all of its outputs?. (I stated for circuits, but I’m happy with the uniform setting too). I’d also be happy if the output space is a bit larger than $\{0,1\}^n$ as long as each $f_n$ is injective.

$\endgroup$
2
  • $\begingroup$ Can you be more precise with what you mean by worst case one-wayness? Is the worst case over the choice of function or over the choice of input? $\endgroup$
    – lamontap
    Commented Oct 3, 2023 at 20:08
  • $\begingroup$ @lamontap I edited the question, I hope it’s a bit more clear now :) $\endgroup$ Commented Oct 3, 2023 at 21:11

1 Answer 1

2
$\begingroup$

The following paper shows that worst-case one-way permutations exist if and only if $\textsf{P} \ne \textsf{UP} \cap \textsf{coUP}$. A problem is in $\textsf{UP}$ if all yes-instances have exactly one witness, and all no-instances have no witnesses. $\textsf{UP}$ is a subset of $\textsf{NP}$.

Christopher M. Homan and Mayur Thakur, One-way permutations and self-witnessing languages.

$\endgroup$
1
  • $\begingroup$ Thanks! This is what I was looking for. $\endgroup$ Commented Oct 3, 2023 at 23:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.