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What does it mean to say a function is length regular and Length preserving? Does any one of them implies the other? Example if any could be useful

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Length-regular: if $x$ and $y$ have the same length, then $f(x)$ and $f(y)$ have the same length.

Length-preserving: $x$ and $f(x)$ have the same length.

Examples:

  • $f(x) = \overline{x}$ (flip every bit in $x$): length-preserving.
  • $f(x) = x \|x$ (concatenate $x$ with itself): length-regular but not length-preserving.
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