TY - JOUR

T1 - Pseudorandom generators for CC0[p] and the Fourier spectrum of low-degree polynomials over finite fields

AU - Lovett, Shachar

AU - Mukhopadhyay, Partha

AU - Shpilka, Amir

N1 - Funding Information:
The authors are supported by the following grants: S.L. is supported by NSF grant DMS-0835373; P.M. is supported by an Aly Kaufman Fellowship and by the Israel Science Foundation (grant 439/06); A.S. is supported by the Israel Science Foundation (grant 339/10). An extended abstract of this paper has appeared as Lovett et al. (2010). We thank Mahdi Cheraghchi for pointing out a small inaccuracy in an earlier version of this paper and the anonymous reviewers for a careful reading of the paper and many helpful suggestions.

PY - 2013/12

Y1 - 2013/12

N2 - In this paper, we give the first construction of a pseudorandom generator, with seed length O(log n), for CC0[p], the class of constant-depth circuits with unbounded fan-in MODp gates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ∈ > 0. In fact, we obtain our generator by fooling distributions generated by low-degree polynomials, over Fp, when evaluated on the Boolean cube. This result significantly extends previous constructions that either required a long seed (Luby et al. 1993) or could only fool the distribution generated by linear functions over Fp, when evaluated on the Boolean cube (Lovett et al. 2009; Meka & Zuckerman 2009). En route 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 Fp. Then, for every ∈ > 0, there exists a subset S ⊂ [n], whose size depends only on d and ∈, such that (Formula presented.). 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 Fp. If the distribution of f when applied to uniform zero-one bits is ∈-far.(in statistical distance) from its distribution when applied to biased bits, then for every δ > 0, f can be approximated over zero-one bits, up to error d, by a function of a small number (depending only on ∈, δ and δ) of lower degree polynomials.

AB - In this paper, we give the first construction of a pseudorandom generator, with seed length O(log n), for CC0[p], the class of constant-depth circuits with unbounded fan-in MODp gates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ∈ > 0. In fact, we obtain our generator by fooling distributions generated by low-degree polynomials, over Fp, when evaluated on the Boolean cube. This result significantly extends previous constructions that either required a long seed (Luby et al. 1993) or could only fool the distribution generated by linear functions over Fp, when evaluated on the Boolean cube (Lovett et al. 2009; Meka & Zuckerman 2009). En route 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 Fp. Then, for every ∈ > 0, there exists a subset S ⊂ [n], whose size depends only on d and ∈, such that (Formula presented.). 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 Fp. If the distribution of f when applied to uniform zero-one bits is ∈-far.(in statistical distance) from its distribution when applied to biased bits, then for every δ > 0, f can be approximated over zero-one bits, up to error d, by a function of a small number (depending only on ∈, δ and δ) of lower degree polynomials.

KW - Small-depth circuits

KW - low-degree polynomials

KW - pseudorandom generators

UR - http://www.scopus.com/inward/record.url?scp=84888301501&partnerID=8YFLogxK

U2 - 10.1007/s00037-012-0051-7

DO - 10.1007/s00037-012-0051-7

M3 - מאמר

AN - SCOPUS:84888301501

VL - 22

SP - 679

EP - 725

JO - Computational Complexity

JF - Computational Complexity

SN - 1016-3328

IS - 4

ER -