Pseudorandom generators with optimal seed length for non-boolean poly-size circuits

Sergei Artemenko, Ronen Shaltiel

Research output: Contribution to journalArticlepeer-review


A sampling procedure for a distribution P over {0, 1} is a function C : {0, 1}n → {0, 1} such that the distribution C(Un) (obtained by applying C on the uniform distribution Un) is the "desired distribution" P. Let n > r ≥ ℓ = nΩ(1). An ∈-nb-PRG (defined by Dubrov and Ishai [2006]) is a function G : {0, 1}r → {0, 1}n such that for every C : {0, 1}n → {0, 1} in some class of "interesting sampling procedures," C′(Ur) = C(G(Ur)) is ∈-close to C(Un) in statistical distance. We construct poly-time computable nb-PRGs with r = O(ℓ) for poly-size circuits relying on the assumption that there exists β > 0 and a problem L in E = DTIME(2O(n)) such that for every large enough n, nondeterministic circuits of size 2βn that have NP-gates cannot solve L on inputs of length n. This assumption is a scaled nonuniform analog of (the widely believed) EXP ≠ ΣP2, and similar assumptions appear in various contexts in derandomization. Previous nb-PRGs of Dubrov and Ishai have r = Ω(ℓ2) and are based on very strong cryptographic assumptions or, alternatively, on nonstandard assumptions regarding incompressibility of functions on random inputs. When restricting to poly-size circuits C : {0, 1}n → {0, 1} with Shannon entropy H(C(Un)) ≤ k, for ℓ > k = nΩ(1), our nb-PRGs have r = O(k). The nb-PRGs of Dubrov and Ishai use seed length r = Ω(k2) and require that the probability distribution of C(Un) is efficiently computable. Our nb-PRGs follow from a notion of "conditional PRGs," which may be of independent interest. These are PRGs where G(Ur) remains pseudorandom even when conditioned on a "large" event {A(G(Ur)) = 1}, for an arbitrary poly-size circuit A. A related notion was considered by Shaltiel and Umans [2005] in a different setting, and our proofs use ideas from that paper, as well as ideas of Dubrov and Ishai. We also give an unconditional construction of poly-time computable nb-PRGs for poly(n)-size, depth d circuits C : {0, 1}n → {0, 1} with r = O(ℓ · logd+O(1)n). This improves upon the previous work of Dubrov and Ishai that has r ≥ ℓ2. This result follows by adapting a recent PRG construction of Trevisan and Xue [2013] to the case of nb-PRGs. We also show that this PRG can be implemented by a uniform family of constant-depth circuits with slightly increased seed length.

Original languageEnglish
Pages (from-to)6:1-6:26
JournalACM Transactions on Computation Theory
Issue number2
StatePublished - Apr 2017

Bibliographical note

Funding Information:
This research was supported by BSF grant 2010120, ISF grant 864/11, and ERC starting grant 279559. A preliminary version of this article appeared in STOC 2014.

Publisher Copyright:
© 2017 ACM.


  • Hardness versus randomness
  • Pseudorandom generators
  • Pseudorandomness
  • Randomness complexity of sampling

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics


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