We prove the existence of a poly(n,m)-time computable pseudorandom generator which “1/poly(n,m)-fools” DNFs with n variables and m terms, and has seed length O(log2 nm ·loglognm). Previously, the best pseudorandom generator for depth-2 circuits had seed length O(log3 nm), and was due to Bazzi (FOCS 2007).

It follows from our proof that a \(1/m^{\tilde O(\log mn)}\)-biased distribution 1/poly(nm)-fools DNFs with m terms and n variables. For inverse polynomial distinguishing probability this is nearly tight because we show that for every m,δ there is a 1/m Ω(log1/δ)-biased distribution X and a DNF φ with m terms such that φ is not δ-fooled by X.

For the case of read-once DNFs, we show that seed length O(logmn ·log1/δ) suffices, which is an improvement for large δ.

It also follows from our proof that a 1/m O(log1/δ)-biased distribution δ-fools all read-once DNF with m terms. We show that this result too is nearly tight, by constructing a \(1/m^{\tilde \Omega(\log 1/\delta)}\)-biased distribution that does not δ-fool a certain m-term read-once DNF.


DNF pseudorandom generators small bias spaces 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Anindya De
    • 1
  • Omid Etesami
    • 1
  • Luca Trevisan
    • 2
  • Madhur Tulsiani
    • 3
  1. 1.University of California at Berkeley 
  2. 2.University of California at Berkeley and Stanford University 
  3. 3.Institute for Advanced Study, Princeton 

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