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Let $\mathcal P$ be the set of permutations in $F_2^n$. I am interested in the circuit complexity of such functions in $AC^k[2]$ setup. What are the relevant upper and lower bounds in this context?

We know the class of permutations, which can be represented as a relabelling of input bits, is in $AC^0[2]$. I am interested in identifying the other classes of permutations, which are realisable in $AC^1[2]$?

Any pointer to suitable references will also suffice?

Thanks in advance

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    $\begingroup$ Err, just draw the wires and you get permutations of input bits even $\mathrm{NC}^0$? Maybe I misunderstand something. $\endgroup$ Commented Jun 1, 2022 at 5:57
  • $\begingroup$ Affine permutations are $\mathrm{AC}^0[2]$. More generally, this holds for all permutations given by constant-degree polynomials. $\endgroup$ Commented Jun 1, 2022 at 11:11
  • $\begingroup$ @VilleSalo Yes, just as the OP says, permutations of input bits are an example of permutations of very low complexity. So what’s the problem with that? $\endgroup$ Commented Jun 1, 2022 at 11:14
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    $\begingroup$ Already NC$^0$ permutations are quite interesting, they can be P-complete to invert csc.kth.se/~johanh/onewaync0.pdf . $\endgroup$ Commented Jun 1, 2022 at 11:23
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    $\begingroup$ Just express each polynomial as a sum of monomials. The outer sum is a parity gate, the monomials are an AND-gate each, and since the degree is constant, the number of monomials is only polynomial. (So I don't really need the polynomials to have constant degree; the only relevant property is that the polynomials have only polynomially many monomials.) $\endgroup$ Commented Jun 7, 2022 at 19:44

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