I was going thorough this paper: (P=BPP unless E has sub-exponential circuits: https://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/IW97/proc.pdf) and I am somewhat struggling with some basics from the paper that I quote below:
Definition 1: Let $m$ and $l$ be positive integers. Let $f:\{{0,1}\}^m→\{{0,1}\}^l$. $S(f)$, the worst-case circuit complexity of $f$, is the minimum number of gates for a circuit $C$ with $C(x)=f(x)$ for every $x∈\{{0,1}\}^m$.
2.2 Hardness versus Randomness: For this section only, we use $f,g,h$ to denote a Boolean function on strings of all lengths, and by $f_n,g_n,h_n$ their segments on n-bit strings. Also, let $E=DTIME(2^{O(n)})$.
Theorem 2: If there is a Boolean function $f∈E$ with $S(f_n )=2^{Ω(n)}$ then $P=BPP$.
Query 1: I am assuming the boolean function is given by its truth table of size $2^m.l$ with all possible entries. I am unclear what it means by this truth table (denoted by $f$) $f∈E/DTIME(2^{O(n)})$? How do you assign the truth table its runtime without having some circuit represent it? I am clear about the fact that $S(f)$ represents the size(i.e. the number of gates in) the smallest possible circuit that represents/calculates $f$.
Query 2: Can someone please explain and simplify the theorem ("if...then") part? I am clear about the complexity classes $P$ and $BPP$, but I am struggling with the condition?