Pseudorandom generators for CC 0 [p] and the Fourier spectrum of low-degree polynomials over finite fields Shachar Lovett ∗ Partha Mukhopadhyay † Amir Shpilka ‡ March 7, 2010 Abstract In this paper we give the first construction of a pseudorandom generator, with seed length O(log n), for CC 0 [p], the class of constant-depth circuits with unbounded fan-in MOD p gates, for some prime p. More accurately, the seed length of our generator is O(log n) for any con- stant error ǫ> 0. In fact, we obtain our generator by fooling distributions generated by low degree polynomials, over F p , when evaluated on the Boolean cube. This result significantly ex- tends previous constructions that either required a long seed [LVW93] or that could only fool the distribution generated by linear functions over F p , when evaluated on the Boolean cube [LRTV09, MZ09]. Enroute of constructing our PRG, we prove two structural results for low degree polynomials over finite fields that can be of independent interest. 1. Let f be an n-variate degree d polynomial over F p . Then, for every ǫ> 0 there exists a subset S ⊂ [n], whose size depends only on d and ǫ, such that ∑ α∈F n p :α=0,α S =0 | ˆ f (α)| 2 ≤ ǫ. Namely, there is a constant size subset S such that the total weight of the nonzero Fourier coefficients that do not involve any variable from S is small. 2. Let f be an n-variate degree d polynomial over F p . If the distribution of f when applied to uniform zero-one bits is ǫ-far (in statistical distance) from its distribution when applied to biased bits, then for every δ> 0, f can be approximated, up to error δ, by a function of a small number (depending only on ǫ, δ and d) of lower degree polynomials. ∗ The Weizmann Institute of Science, shachar.lovett@weizmann.ac.il. Research supported by the Israel Science Foundation (grant 1300/05). † Technion - Israel Institute of Technology, partha@cs.technion.ac.il. Research supported in part at the Tech- nion by an Aly Kaufman Fellowship. ‡ Technion - Israel Institute ofTechnology, shpilka@cs.technion.ac.il. Research supported by the Israel Science Foundation (grant 439/06). 1