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I have an intuitive idea of this, but I am not sure if I am formally interpreting it correctly. In the scenario I am considering, each party is identified by a sequence of $n$ bits and I have $2^{n}$ parties. If I say that an adversary can control a polynomial number of parties, what does it mean formally? Is it with respect to $n$, $2^{n}$ or what? How do you explicit it as a function?

My idea is that, since the number of parties is exponential with respect to $n$ (in particular, it is $2^{n}$), then, saying that the adversary controls a polynomial number of parties means that it controls a number of parties that is polynomial with respect to $n$. It can be $2 \cdot n$, $n^{2}$ or any polynomial function of $n$ (it should be possible to express it in general as $poly(n)$).

Is this correct?

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    $\begingroup$ Usually in complexity theoretical cryptography when we say "polynomial in x" we mean "polynomial in the length of x" which means "polynomial in $1^n$" in cryptography. $\endgroup$
    – SEJPM
    May 17, 2020 at 14:40
  • $\begingroup$ Thanks a lot. It seems to be equivalent to what I assumed then. $\endgroup$
    – Lorenzo
    May 17, 2020 at 14:55

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