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Next-Block Pseudoentropy ! PRG

Next-Block Pseudoentropy ! PRG. g(x 1 ,h 1 )=. g(x 2 ,h 2 )=. …. f(x 2 ). f( x t ). f(x 1 ). h. h. h. h( x 2 ). h( x t ). h( x 1 ). g( x t ,h t )=. …. extractors. G(x 1 ,h 1 …, x t ,h t ). n+|h|+ 1 bits of next-block pseudoentropy.

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Next-Block Pseudoentropy ! PRG

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  1. Next-Block Pseudoentropy ! PRG g(x1,h1)= g(x2,h2)= … f(x2) f(xt) f(x1) h h h h(x2) h(xt) h(x1) g(xt,ht)= … extractors G(x1,h1…,xt,ht) n+|h|+ 1 bits of next-block pseudoentropy

  2. Next-Block Pseudoentropy ! PRG • Distinguisher for G ) next-bit predictor for G )(hybrid) next-bit predictor for g )gdoes not have high next-bit pseudoentropy • Seed length O(n3), but construction is (highly) non-uniform • “Entropy equalization” )uniform construction with seed length O(n4)

  3. Entropy Equalization Task: Given X=(X1…Xn) with next-block-entropy k, construct X’ =(X’1…X’n’) for which  Y’=(Y’1…Y’n’) with • 8i (X’1…X’i-1,X’i)≈C (X’1…X’i-1,Y’i) • 8i H(Y’i|X’1…X’i-1) = k/n - ± X’ = (X(1)I,X(1)I+1…X(1)n,X(2)1, …X(t)I-n)andY’ = (Y(1)I,Y(1)j+1…Y(1)n,Y(2)1, …Y(t)I-n) 8i H(Y’i|X’1…X’i-1) = k/n - k/(t-1)n … … X(1)1 X(1)1 X(t)1 X(1)2 X(t)n X(2)2 … … X(1)n X(2)n X(t)n n-j j

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