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Parity of the sum of pseudorandom bits over a non-sparse set of inputs

You can't say anything useful. Let $E$ be a pseudorandom permutation on $\{0,1\}^m$ with $n$-bit key, $m$ be large (at least twice the security parameter), $S=\{0,1\}^m$, and define $F$ by $$F(v,w) = ...
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