Better Pseudorandom Generators from Milder Pseudorandom Restrictions

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“…The construction of the generator in Theorem 1.1 is essentially the same as the generator of Gopalan et al [18] for read-once CNF formulas, which was used by Reingold et al [36] for read-once, oblivious, permutation branching programs. In our analysis, we give a new bound on the Fourier mass of oblivious, read-once, width-3 branching programs.…”

“…The analysis of Gopalan et al [18] does not rely on the order in which the output bits are read, and the previously mentioned results of Reingold, Steinke, and Vadhan [36] use Fourier analysis of regular, read-once, oblivious branching programs to show that the generator of Gopalan et al fools unordered, read-once, oblivious permutation branching programs.…”

“…A bound on the Fourier growth of f , or the rate at which L k ( 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. More precisely, they show that for any class C of functions computed by read-once, oblivious, branching programs that is closed under restrictions and decompositions and satisfies L k ( f ) ≤ poly(n) · c k for every k and f ∈ C, there is a pseudorandom generator with seed length O(c · log 2 n) that fools every f ∈ C. They then bound the Fourier growth of read-once, oblivious, permutation branching programs (and the even more general model of read-once, oblivious, "regular" branching programs, where each layer between consecutive state spaces is a regular bipartite graph) to obtain a pseudorandom generator for read-once, oblivious, permutation branching programs.…”

“…Recently, a new approach for constructing pseudorandom generators has been suggested in the paper of Gopalan et al [18]; they constructed pseudorandom generators for read-once CNF formulas and combinatorial rectangles, and hitting set generators for ordered width-3 branching programs, all having | E [P(G(U ))] − E [P(U n )] | ≤ ε, where U and U n are uniform on {0, 1} and on {0, 1} n , respectively.…”

“…1 Despite intensive study, this is the best known seed length for ordered branching programs even of width 3, but a variety of results have shown improvements for restricted classes of ordered branching programs such as width-2 programs [39,5], and "regular" or "permutation" ordered branching programs (of constant width) [9,10,28,14,45]. 2 For width 3, hitting set generators (a relaxation of pseudorandom generators) have been constructed [42,18]. The vast majority of these results are based on Nisan's original generator or its variants by Impagliazzo, Nisan, and Wigderson [25] and Nisan and Zuckerman [34], and adhere to a paradigm that seems unlikely to yield generators against general logspace computations with seed length better than log 1.99 n (see [10]).…”