kb autocommit

content/posts/KBhtopics_efficient_derandomization.md

@@ -31,3 +31,16 @@ G : \qty {0,1}^{\log n}  \to  \qty {0,1}^{n}
 Naively doing this is certainty doesn't result in a uniform output distribution; because \\(G\\) is wildly not [surjective]({{< relref "KBhsurjectivity.md" >}}) --- there are \\(2^{\log n} = n\\) possible input values, and \\(2^{n}\\) possible outputs.
 
 We want \\(M\\) to not be able to tell the difference, in particular meaning we want \\(G\\) to be scrambly enough o that \\(M\\) can't tell the diffidence between these two pictures.
+
+We can't technically do this but we need to construct \\(G\\) mapping \\(G:  \qty {0,1}^{\log n} \to  \qty {0,1}^{n}\\) with...
+
+1.  \\(G\\) is computer poly \\(\qty(n)\\) (in particular \\(n^{3}\\) time---i.e. more time than the underlying turing machine)
+2.  for all randomized \\(n^{2}\\) time, turing machine, \\(\forall  x \in \qty {01}^{\*}\\), we desire:
+
+\begin{equation}
+Pr\_{r \sim \qty {0,1}^{n}}, \qty [M \qty(x,r), \text{accept}] = Pr\qty [M\qty(x, G\qty(s)))] \pm 0.1
+\end{equation}
+
+i.e. it's "fooled" by \\(G\\)
+
+**insight**: randomness derives from the inability for \\(M\\) to compute harder than \\(n^{2}\\). "Randomness is in the eye of the beholder."