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Computational Indistinguishability “To suppose two things indiscernible is to suppose the same thing under two different names” Gottfried Wilhelm Leibniz (1646-1716) Def: Let X,Y be dist. over {0,1} n. We say X,Y are (T, )-indistinguishable if 8 T-sized circuit C:{0,1} n {0,1}, | Pr[ C(X)=1 ] – Pr[ C(Y) =1] | < Fact: X and Y are -close if and only if they are (1, )-indistinguishable. Lemma: 8 n, 9 X and Y that are (2 n/10,2 -n/10 )-indistinguishable, but statistically 0.9-far. Recall: The statistical distance between X and Y is: X,Y) = ½ w | Pr X [w] – Pr W [y] | = max | Pr X [T] – Pr Y [T] | Tµ {0,1} n Notation: X T, Y
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Properties of Indistinguishability Def: Let X,Y be dist. over {0,1} n. We say X,Y are (T, )-indistinguishable if 8 T-sized circuit C:{0,1} n {0,1}, | Pr[ C(X)=1 ] – Pr[ C(Y) =1] | < Lemma (Transitivity): X T, Y, Y T, Z ) X 10T, 2 Z Notation: X T, Y Lemma (Concatenation): X T, Y, X’ T, Y’ ) (X,X’) 10T, 2 (Y,Y’)
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Pseudo-Randomness That is, 8 T-sized circuit C:{0,1} n {0,1}, | Pr[ C(X)=1 ] – Pr[ C(U n ) =1] | < Def: Let X be dist. over {0,1} n. We say X is (T, )-pseudorandom if X is (T, )-indistinguishable from U n. We know: 9 (2 n/10,2 -n/10 )-pseudorandom X s.t. X is 0.9-far from U n. Def: Let G:{0,1} s {0,1} n be s.t. s<n. G(¢) is (T, )-pseudorandom generator if g(U s ) is (T, )-pseudorandom. snGLemma: 9 (2 n/10,2 -n/10 )-PRG G with n=2 s/10.
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Pseudo-Randomness Def: Let G:{0,1} s {0,1} n be s.t. s<n. G(¢) is (T, )-pseudorandom generator if g(U s ) is (T, )-pseudorandom. snG Def: Let { G n } be function family such that G n :{0,1} s {0,1} n, for s=s(n)<n. { G n } is a pseudorandom generator if 9 super-poly h(¢) such that 8 n, G n is a ( h(n), 1/h(n)-PRG Notation: We let G = [ n G n. That is, 8x2 {0,1} *, G(x)=G |x| (x). We know: There exists a PRG G. Million $ question: Is there an efficiently computable PRG G? Lemma: If NPµ P/poly then @ PRG G.
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