It is known that (worst-case) one-way permutations exist if and only if $P\ne UP \cap coUP$. Almost all candidates that I know are based on hard number theortic problems. I came across a combinatorial construction of one-way permutations based on quasigroup string transformations.

What other combinatorial candidates for one-way permutations are known?

  • $\begingroup$ Are you interested in trapdoor permutation families over products of fixed-size $\hspace{1.81 in}$ (at most 8 elements) finite fields? ​ ​ $\endgroup$ – user6973 Sep 22 '16 at 17:46
  • $\begingroup$ @RickyDemer No, just basic worst-case one-way permutation would be fine. $\endgroup$ – Mohammad Al-Turkistany Sep 22 '16 at 19:20
  • $\begingroup$ Hmm, that's harder. ​ ​ $\endgroup$ – user6973 Sep 22 '16 at 19:47
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    $\begingroup$ Relevant: eprint 2015/752: eprint.iacr.org/2015/752 $\endgroup$ – D.W. Sep 23 '16 at 5:34
  • $\begingroup$ Is there a specific use-case you have in mind for one-way permutations? ​ ​ $\endgroup$ – user6973 Sep 25 '16 at 6:32

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