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lec9 - CS151 Complexity Theory Lecture 9 Derandomization...

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CS151 Complexity Theory Lecture 9 April 26, 2011
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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
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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
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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”)
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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
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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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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…
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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 )
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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
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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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April 26, 2011 11 Hardness vs. randomness BMY pseudo-random generator: one generator fooling all poly-size bounds one-way-permutation is hard function implies hard function in NP coNP
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  • Fall '09
  • Analysis of algorithms, Computational complexity theory, Error detection and correction, Automatic repeat request, output length, seed length

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