Research output per year
Research output per year
Sergei Artemenko, Ronen Shaltiel
Research output: Contribution to journal › Article › peer-review
A sampling procedure for a distribution P over {0, 1}^{ℓ} is a function C : {0, 1}^{n} → {0, 1}^{ℓ} such that the distribution C(U_{n}) (obtained by applying C on the uniform distribution U_{n}) is the "desired distribution" P. Let n > r ≥ ℓ = n^{Ω(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}^{ℓ} in some class of "interesting sampling procedures," C′(U_{r}) = C(G(U_{r})) is ∈-close to C(U_{n}) 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(2^{O(n)}) such that for every large enough n, nondeterministic circuits of size 2^{βn} that have NP-gates cannot solve L on inputs of length n. This assumption is a scaled nonuniform analog of (the widely believed) EXP ≠ Σ^{P}_{2}, and similar assumptions appear in various contexts in derandomization. Previous nb-PRGs of Dubrov and Ishai have r = Ω(ℓ^{2}) 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}^{ℓ} with Shannon entropy H(C(U_{n})) ≤ k, for ℓ > k = n^{Ω(1)}, our nb-PRGs have r = O(k). The nb-PRGs of Dubrov and Ishai use seed length r = Ω(k^{2}) and require that the probability distribution of C(U_{n}) is efficiently computable. Our nb-PRGs follow from a notion of "conditional PRGs," which may be of independent interest. These are PRGs where G(U_{r}) remains pseudorandom even when conditioned on a "large" event {A(G(U_{r})) = 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}^{ℓ} with r = O(ℓ · log^{d+O(1)}n). This improves upon the previous work of Dubrov and Ishai that has r ≥ ℓ^{2}. 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 language | English |
---|---|
Pages (from-to) | 6:1-6:26 |
Journal | ACM Transactions on Computation Theory |
Volume | 9 |
Issue number | 2 |
DOIs | |
State | Published - Apr 2017 |
Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review