I am given an oracle $A$ that takes input samples from two distributions $\chi_1$ and $\chi_2$.

Suppose we have $Pr_{x \sim \chi_1}[A(x) = 1] = p_1$ and $Pr_{x \sim \chi_2}[A(x) = 1] = p_2$, where $p_1 \neq p_2$.

In general, how can we use $A$ to construct a distinguisher to determine whether the input samples are from $\chi_1$ or $\chi_2$? And how good is the running time of such distinguisher?

Thank you very much! :)

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    $\begingroup$ What have you tried? Where did you get stuck? Do you expect us to solve your question for you? $\endgroup$ May 5, 2013 at 5:16
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    $\begingroup$ Cross-posted on cstheory: cstheory.stackexchange.com/questions/17518/…. Don't do that. Your question there will be closed. $\endgroup$ May 5, 2013 at 5:18

1 Answer 1


Hint: Run $A$ on lots of samples $x_1,\ldots,x_n$ and compute the average $(A(x_1)+\cdots+A(x_n))/n$.


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