Suppose $X$ is a message which takes values on the set $\{x_1, \dots, x_m\}$ with probability distribution $P_X$. We transmit the message $X$ over  the channel $P_{Y|X}$ which outputs $Y$ taking values on finite alphabet $\mathcal{Y}$. Suppose we take $n$ independent samples from channel output; namely, $Y^n=(Y_1, Y_2, \dots, Y_n)$ with distribution $\prod_{i=1}^nP_{Y|X}(y_k|x)$ where $x$ is the message sent over the channel.Hence, we have $m$ distributions $\{P_{Y^n|X}(\cdot|x_1), \dots, P_{Y^n|X}(\cdot|x_m)\}$. I am looking for a deterministic randomness extractor which works for all of these $m$ distributions, that is, a randomness $Ext:\mathcal{Y^n}$ such that for given $Y^n$ distributed as one of these $m$ distribution $Ext(Y^n)$ is $\epsilon-$close to uniform distribution over $\{0,1\}^r$ (in total variation distance sense). What is the largest random bits that we can extract, i.e., the largest $r$? Is that $\min_{x}H_{\infty}(P_{Y^n|X}(\cdot|x))$?The motivation of this comes from privacy.  
deterministic randomness extractor and privacy
cr.crypto security;randomness;it.information theory;extractors;privacy