Presentation is loading. Please wait.

Presentation is loading. Please wait.

Computational Indistinguishability “To suppose two things indiscernible is to suppose the same thing under two different names” Gottfried Wilhelm Leibniz.

Similar presentations


Presentation on theme: "Computational Indistinguishability “To suppose two things indiscernible is to suppose the same thing under two different names” Gottfried Wilhelm Leibniz."— Presentation transcript:

1 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

2 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’)

3 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.

4 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.


Download ppt "Computational Indistinguishability “To suppose two things indiscernible is to suppose the same thing under two different names” Gottfried Wilhelm Leibniz."

Similar presentations


Ads by Google