# What does it mean exactly that an adversary can control a polynomial number of parties?

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?

• 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. May 17, 2020 at 14:40
• Thanks a lot. It seems to be equivalent to what I assumed then. May 17, 2020 at 14:55