Machine Theory

A primer on pseudorandom generators by Oded Goldreich

By Oded Goldreich

A clean examine the query of randomness used to be taken within the concept of computing: A distribution is pseudorandom if it can't be exceptional from the uniform distribution by way of any effective method. This paradigm, initially associating effective tactics with polynomial-time algorithms, has been utilized with admire to quite a few traditional sessions of distinguishing systems. The ensuing conception of pseudorandomness is suitable to technology at huge and is heavily concerning imperative parts of computing device technology, resembling algorithmic layout, complexity idea, and cryptography. This primer surveys the speculation of pseudorandomness, beginning with the overall paradigm, and discussing quite a few incarnations whereas emphasizing the case of general-purpose pseudorandom turbines (withstanding any polynomial-time distinguisher). extra issues contain the "derandomization" of arbitrary probabilistic polynomial-time algorithms, pseudorandom turbines withstanding space-bounded distinguishers, and a number of other typical notions of special-purpose pseudorandom turbines. The primer assumes uncomplicated familiarity with the proposal of effective algorithms and with effortless chance conception, yet presents a simple advent to all notions which are really used. accordingly, the primer is basically self-contained, even though the reader is now and then said different assets for extra element

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In particular, for every polynomial-time constructible ensemble {Xn }n∈N , every Boolean function f ∈ BPP, and every ε > 0, there exists a randomized algorithm A′ of randomness complexity rε (n) = nε such that the probability that A′ (Xn ) = f (Xn ) is negligible. A corresponding deterministic (exp(rε )- 26 CHAPTER 2. GENERAL-PURPOSE PSEUDORANDOM GENERATORS domization” of a complexity class such as BPP. To obtain such results, we need a stronger notion of pseudorandom generators, presented next.

Def Guideline: If D distinguishes the latter ensembles, then D′ such that D′ (z) = D(A(z)) distinguishes the former. 4, show that the conclusion may not hold when A is not computationally bounded. That is, show that there exists computationally indistinguishable ensembles, {Xn }n∈N and {Yn }n∈N , and an exponentialtime algorithm A such that {A(Xn )}n∈N and {A(Yn )}n∈N are not computationally indistinguishable. Guideline: For any pair of ensembles {Xn }n∈N and {Yn }n∈N , consider the Boolean function f such that f (z) = 1 if and only if Pr[Xn = z] > Pr[Yn = z].

2, we seek canonical derandomizers with a stretch that is as large as possible. , it must hold that ℓ(k) = O(2k )), because there exists a circuit of size O(2k · ℓ(k)) that violates Eq. 2) whereas for ℓ(k) = ω(2k ) it holds that O(2k · ℓ(k)) < ℓ(k)2 . Thus, our goal is to construct a canonical derandomizer with stretch ℓ(k) = 2Ω(k) . 3 (derandomization of BPP, revisited): If there exists a canonical derandomizer of stretch ℓ(k) = 2Ω(k) , then BPP = P. 2, we get BPtime(t) ⊆ Dtime(T ), where T (n) = −1 poly(2ℓ (t(n)) · t(n)) = poly(t(n)).

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