Pseudorandomness from Shrinkage

Abstract

One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer a quantitative loss in parameters, and hence do not give nontrivial implications for models where we don't know superpolynomial lower bounds but do know lower bounds of a fixed polynomial. We show that when such lower bounds are proved using random restrictions, we can construct PRGs which are essentially best possible without in turn improving the lower bounds. More specifically, say that a circuit family has shrinkage exponent Γ if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Γ+o(1)</sup> . Our PRG uses a seed of length s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/(Γ+1)+o(1)</sup> to fool circuits in the family of size s. By using this generic construction, we get PRGs with polynomially small error for the following classes of circuits of size s and with the following seed lengths: 1) For de Morgan formulas, seed length s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/3+o(1)</sup> ; 2) For formulas over an arbitrary basis, seed length s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2+o(1)</sup> ; 3) For read-once de Morgan formulas, seed length s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">.234...</sup> ; 4) For branching programs of size s, seed length s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2+o(1)</sup> . The previous best PRGs known for these classes used seeds of length bigger than n/2 to output n bits, and worked only when the size s = O(n) [1].