Supposing Alice and Bob share $\rho$-correlated sequences in $\{0,1\}^n$, what coding theory based schemes are available for Alice and Bod to extract sequences $A,B\in\{0,1\}^n$ respectively such that $A,B$ agree on $k(n,\rho)$ bits for cases (I am not worried about secrecy):
$(1)$ Alice and Bob communicate.
$(2)$ Alice and Bob do not communicate.
Assume Alice and Bob exchange $m(n,\rho)$ bits in total to agree on $k(n,\rho)$ bits.
What is the min rate $\lim_{n\rightarrow\infty}\frac{m(n,\rho)}{k(n,\rho)}$?
Rather more generally, completely, what is explicit formula for the curve $\lim_{n\rightarrow\infty}(m(n,\rho),k(n,\rho))\subseteq\Bbb R^2$?