A primer on pseudorandom generators by Oded Goldreich

By Oded Goldreich

A clean examine the query of randomness was once taken within the conception of computing: A distribution is pseudorandom if it can't be distinct from the uniform distribution via any effective approach. This paradigm, initially associating effective systems with polynomial-time algorithms, has been utilized with recognize to quite a few average sessions of distinguishing systems. The ensuing idea of pseudorandomness is appropriate to technology at huge and is heavily on the topic of valuable parts of desktop technology, reminiscent of algorithmic layout, complexity idea, and cryptography. This primer surveys the speculation of pseudorandomness, beginning with the overall paradigm, and discussing a number of incarnations whereas emphasizing the case of general-purpose pseudorandom turbines (withstanding any polynomial-time distinguisher). extra themes comprise the "derandomization" of arbitrary probabilistic polynomial-time algorithms, pseudorandom turbines withstanding space-bounded distinguishers, and several other normal notions of special-purpose pseudorandom turbines. The primer assumes uncomplicated familiarity with the inspiration of effective algorithms and with undemanding chance idea, yet offers a simple creation to all notions which are really used. accordingly, the primer is largely self-contained, even supposing the reader is now and then spoke of different assets for extra aspect

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Using 1/6 as the “threshold distinguishing gap” (in Eq. , Pr[Dk (G(Uk )) = 1] < 1/2). As we shall see, this suffices for a derandomization of BPtime(t) in −1 time T , where T (n) = poly(2ℓ (t(n)) · t(n)) (and we use a seed of length k = ℓ−1 (t(n))). 2. )4 Observe that the complexity of the resulting deterministic proce−1 dure is dominated by the 2k = 2ℓ (t(|x|)) invocations of AG (x, s) = A(x, G(s)), where −1 s ∈ {0, 1}k , and each of these invocations takes time poly(2ℓ (t(|x|) )+t(|x|). Thus, on −1 input an n-bit long string, the deterministic procedure runs in time poly(2ℓ (t(n)) · t(n)).

Recall that saying that a function f is one-way means that given a typical y (in the range of f ) it is infeasible to find a preimage of y under f . This does not mean that it is infeasible to find partial information about the preimage(s) of y under f . , given a one-way function f consider the function f ′ defined def by f ′ (x, r) = (f (x), r), for every |x| = |r|). We note that hiding partial information (about the function’s preimage) plays an important role in the construction of pseudorandom generators (as well as in other advanced constructs).

Indeed, the implication is due to Eq. 1), when applied to the circuit Cx (r) = A(x, r) (which has size at most |r|2 ). The goal. 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.

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