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I am by no means a security expert, but I am curious whether functions that don't produce different outputs when raised to the power of N (without any psuedo-randomness) exist, that are as strong as "bcrypt" or "scrypt"?

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Could you make your requirements a bit clearer? Do you mean that $f(f(x)) = f(x)$? –  PaĆ­lo Ebermann Jan 21 '14 at 22:10

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