lec9 - Derandomization Goal try to simulate BPP in...

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1 CS151 Complexity Theory Lecture 9 April 26, 2011 April 26, 2011 2 Derandomization • Goal : try to simulate BPP in subexponential time (or better) • use Pseudo-Random Generator (PRG): • often: PRG “good” if it passes (ad-hoc) statistical tests seed output string G t bits m bits April 26, 2011 3 Derandomization • ad-hoc tests not good enough to prove BPP has non-trivial simulations • Our requirements: – G is efficiently computable – “ stretches t bits into m bits – “ fools ” small circuits: for all circuits C of size at most s : |Pr y [C(y) = 1] – Pr z [C(G(z)) = 1]| ≤ ε April 26, 2011 4 Simulating BPP using PRGs • Recall: L BPP implies exists p.p.t.TM M x L Pr y [M(x,y) accepts] ≥ 2/3 x L Pr y [M(x,y) rejects] ≥ 2/3 • given an input x: – convert M into circuit C(x, y) – simplification: pad y so that |C| = |y| = m • hardwire input x to get circuit C x Pr y [C x (y) = 1] ≥ 2/3 (“yes”) Pr y [C x (y) = 1] ≤ 1/3 (“no”) April 26, 2011 5 Simulating BPP using PRGs • Use a PRG G with – output length m – seed length t « m – error ε < 1/6 – fooling size s = m • Compute Pr z [C x (G(z)) = 1] exactly – evaluate C x (G(z)) on every seed z {0,1} t • running time (O(m)+(time for G)) 2 t April 26, 2011 6 Simulating BPP using PRGs • knowing Pr z [C x (G(z)) = 1] , can distinguish between two cases: 0 1/3 1/2 2/3 1 “yes”: ε 0 1/3 1/2 2/3 1 “no”: ε
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2 April 26, 2011 7 Hardness vs. randomness • We have shown: If one-way permutations exist then BPP δ>0 TIME(2 n δ ) ( EXP • simulation is better than brute force, but just barely • stronger assumptions on difficulty of inverting OWF lead to better simulations… April 26, 2011 8 Hardness vs. randomness • We will show: If E requires exponential size circuits then BPP = P by building a different generator from different assumptions. E = k DTIME(2 kn ) April 26, 2011 9 Hardness vs. randomness • BMY: for every δ > 0, G δ is a PRG with seed length t = m δ output length m error ε < 1/m d (all d) fooling size s = m e (all e) running time m c • running time of simulation dominated by 2 t April 26, 2011 10 Hardness vs. randomness • To get BPP = P , would need t = O(log m) • BMY building block is one-way- permutation: f:{0,1} t → {0,1} t • required to fool circuits of size m e (all e) • with these settings a circuit has time to invert f by brute force! can’t get BPP = P with this type of PRG
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lec9 - Derandomization Goal try to simulate BPP in...

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