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I remember reading somewhere that (under certain reasonable assumptions) a Boolean circuit with many inputs and outputs (assume equal number for now) chosen at random will be a one-way function with high probability. Is this true?

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    $\begingroup$ It depends on how you define random circuit. If you use $n$ block inputs, $m$ NAND gates each with 2 inputs and one output, with the block's outputs these $m$ gate outputs, and randomly connect each of the $2m$ gate inputs to one of the $n+m$ block inputs and gate outputs, you often end up with something that is not a function (two NAND gates can make an RS latch). If you add some rules preventing this, that's a function but not a one-way function. You need to hide many of the outputs to hope for that. I know no quantitative result (hence this is not an answer) [fixed] $\endgroup$ – fgrieu Mar 1 '16 at 11:30
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The claim that you remember is to good to be true. It is yet unknown (because the question P ?= NP is still open) whether there are one-way functions. If the claim you remember was true, it would prove the existence of one-way functions.

P.S. While P = NP excludes the existence of one-way functions, even with P != NP the existence of one-way functions is still an open problem.

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    $\begingroup$ This is mathematically correct; but from an applied cryptography standpoint, there are one-way functions. $\endgroup$ – fgrieu Mar 1 '16 at 11:32
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    $\begingroup$ I would count "the existence of one-way functions" as a reasonable (even if unproven) assumption. $\endgroup$ – poncho Mar 1 '16 at 14:07
  • $\begingroup$ @fgrieu: I am explicitly referring to the original question. The vaguely remembered claim about some random circuits as stated by the OP would imply the existence of one-way functions. $\endgroup$ – jk - Reinstate Monica Mar 1 '16 at 14:13
  • $\begingroup$ It could be a claim under some widely believed assumption. Although it seems very unlikely for average-case hard OWF, the question is interesting for worst-case hard OWF. $\endgroup$ – Geoffroy Couteau Mar 1 '16 at 14:18
  • $\begingroup$ My question mentioned "reasonable assumptions". That includes "there is a one-way function". $\endgroup$ – Demi Mar 1 '16 at 15:14

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