Lecture 15: Simulating Bpp -high Min-entropy Sources 1 Simulating Bpp Using Sv-sources

It’s easy to check that any specific string of length n has probability at most (1 − δ), hence SV-sources are n log 1 1−δ -sources. Also recall that SV-sources are (k, l) block sources where k = l log 1 1−δ . Suppose δ = log 1 1−δ , then k = δl. Take a hash-based extractor E : {0, 1} × {0, 1} → {0, 1}. Here, our weak-source is a SVsource over the strings of length n and suppose k = n log 1 1−δ is the lower bound for min-entropy of the source. According to construction of hash-based extractors (Lecture 13) d = O(n), d+m = k + d − 2 log 1 ǫ , hence m = k − 2 log 1 ǫ . (ǫ is, as usual, statistical distance of output and uniform distribution).