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.