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1

It's simply due to a hash function acting (within it's block size $n$) as:- You'll notice that there is no possible 'A' output as there was a collision at 'C'. It's a version of the pigeon hole principle meaning that when two birds occupy one hole, you must have an empty hole remaining. Some of the output bins of a pseudo random function (hash extractor) ...


2

I didn't follow all of the question, but let's take a small example. Suppose $X$ is uniformly distributed in $\{1,2,3\}$, so that $\Pr[X = x] = 1/3$ for any $x \in \{1,2,3\}$. If we define \begin{equation*} f(1) = 2, \qquad f(2) = 3, \qquad f(3) = 2, \end{equation*} then we have \begin{align*} \Pr[f(X) = 1] &= 0, \\ \Pr[f(X) = 2] &= \...


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Assuming knowing the entropy in the generation process is meaningful, the relationship between difficulty of brute force guessing an unknown random variable 𝑋 and entropy (or entropies) is quite a delicate one, and depends on the assumptions about the attack scenario. In particular, the direct use of Shannon entropy can give misleading results, and the ...


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