We consider the following problem: for given n,M, produce a sequence X 1,X 2,…,X n of bits that fools every linear test modulo M. We present two constructions of generators for such sequences. For every constant prime power M, the first construction has seed length O M (log(n/ε)), which is optimal up to the hidden constant. (A similar construction was independently discovered by Meka and Zuckerman [MZ]). The second construction works for every M,n, and has seed length O(logn + log(M/ε)log(Mlog(1/ε))).

The problem we study is a generalization of the problem of constructing small bias distributions [NN], which are solutions to the M = 2 case. We note that even for the case M = 3 the best previously known constructions were generators fooling general bounded-space computations, and required O(log2 n) seed length.

For our first construction, we show how to employ recently constructed generators for sequences of elements of ℤ M that fool small-degree polynomials (modulo M). The most interesting technical component of our second construction is a variant of the derandomized graph squaring operation of [RV]. Our generalization handles a product of two distinct graphs with distinct bounds on their expansion. This is then used to produce pseudorandom-walks where each step is taken on a different regular directed graph (rather than pseudorandom walks on a single regular directed graph as in [RTV, RV]).


Hash Function Cayley Graph Seed Length Linear Test Pseudorandom Generator 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Shachar Lovett
    • 1
  • Omer Reingold
    • 1
  • Luca Trevisan
    • 2
  • Salil Vadhan
    • 3
  1. 1.Department of Computer ScienceWeizmann Institute of ScienceRehovotIsrael
  2. 2.Computer Science DivisionUniversity of CaliforniaBerkeleyUSA
  3. 3.School of Engineering and Applied ScienceHarvard UniversityCambridgeUSA

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