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A hard-core predicate of a one-way function $f$ is a predicate b (i.e., a function whose output is a single bit) which is easy to compute (as a function of x) but is hard to compute given $f(x)$.
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Why can't we construct a PRG from a one-way function and hc, but only one-way permutation
From Katz & Lindell's book, theorem 7.19:
Let f be a one way permutation with hard-core predicate hc.
Then the algorithm $G(s)=f(s)||hc(s)$ is a PRG with expansion factor $\ell(n)=n+1$.
But what abo …