Abstract
We present an explicit pseudorandom generator for oblivious, read-once, width-3 branching programs, which can read their input bits in any order. The generator has seed length ˜ O(log 3 n). The previously best known seed length for this model is n 1/2+o(1) due to Impagliazzo, Meka, and Zuckerman (FOCS ’12). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM ’13) for permutation branching programs. The main technical novelty underlying our generator is a new bound on the Fourier growth of width-3, oblivious, read-once branching programs. Specifically, we show that for any f :{0, 1} n !{0, 1} computed by such a branching program, and k2 [n], X s [n]:|s|=k b f[s] n 2 · (O(logn)) k ,
Highlights
1.1 Pseudorandom generators for space-bounded computationA central problem in the theory of pseudorandomness is constructing an “optimal” pseudorandom generator for space-bounded computation—that is, an explicit algorithm that stretches a uniformly random seed of O(log n) bits to n bits that cannot be distinguished from uniform by any log n-space algorithm
A recent line of work [6, 24, 36] has constructed pseudorandom generators for unordered, readonce, oblivious branching programs; none match both the seed length and generality of Nisan’s result
A bound on the Fourier growth of f, or the rate at which Lk( f ) grows with k, was used by Mansour [30] to obtain an improved query algorithm for polynomial-size DNF; the junta approximation results of Friedgut [17] and Bourgain [7] are proved using approximating functions that have slow Fourier growth. This notion turns out to be useful in the analysis of pseudorandom generators as well: Reingold et al [36] show that the generator of Gopalan et al [18] will work if there is a good bound on the Fourier mass of low-order coefficients
Summary
A central problem in the theory of pseudorandomness is constructing an “optimal” pseudorandom generator for space-bounded computation—that is, an explicit algorithm that stretches a uniformly random seed of O(log n) bits to n bits that cannot be distinguished from uniform by any log n-space algorithm. Most previous constructions of pseudorandom generators for space-bounded computations consider ordered branching programs, where the input bits are read in order—that is, π(i) = i. The classic result of Nisan [33] gave a generator stretching O(log n) uniformly random bits to n bits that are pseudorandom against ordered branching programs of polynomial width.. A recent line of work [6, 24, 36] has constructed pseudorandom generators for unordered, readonce, oblivious branching programs (where the bits are fed to the branching program in an arbitrary, fixed order); none match both the seed length and generality of Nisan’s result. Our main result shows that the pseudorandom generator of Gopalan et al with seed length O(log n) fools width-3, read-once, oblivious branching programs. We give a new bound on the Fourier mass of oblivious, read-once, width-3 branching programs
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