Pseudorandomness via the Discrete Fourier Transform

Abstract

We present a new approach to constructing unconditional pseudorandom generators against classes of functions that involve computing a linear function of the inputs. We give an explicit construction of a pseudorandom generator that fools the discrete Fourier transforms of linear functions with seed-length that is nearly logarithmic (up to polyloglog factors) in the input size and the desired error parameter. Our result gives a single pseudorandom generator that fools several important classes of tests computable in logspace that have been considered in the literature, including halfspaces (over general domains), modular tests and combinatorial shapes. For all these classes, our generator is the first that achieves near logarithmic seed-length in both the input length and the error parameter. Getting such a seed-length is a natural challenge in its own right, which needs to be overcome in order to derandomize RL -a central question in complexity theory.Our construction combines ideas from a large body of prior work, ranging from a classical construction of [NN93] to the recent gradually increasing independence paradigm of [KMN11, CRSW13, GMR + 12], while also introducing some novel analytic machinery which might find other applications.

Perfect Lp Sampling in a Data Stream

“…We now show that our algorithm L p Sampler with Fast-Update can be derandomized without affecting the space or time complexity. To do this, we use a combination of Nisan's pseudorandom generator (PRG) [Nis92], and the PRG of Goplan, Kane, and Meka [GKM15]. We begin by introducing Nisan's PRG, which is a deterministic map G : {0, 1} ℓ → {0, 1} T , where T ≫ ℓ (here we think of T = poly(n) and ℓ = O(log 2 (n))).…”

“…Using extensions of the techniques in that paper, we demonstrate that the same PRG with a smaller precision ǫ can be used to fool functions of more half-spaces. We now introduce the main result of [GKM15]. Let C 1 = {c ∈ C | |c| ≤ 1}.…”

“…Theorem 4 (Theorem 1.1 [GKM15]). There is a PRG G : {0, 1} ℓ → [m] n that fools all (m, n)-Fourier shapes f with error ǫ using a seed of length ℓ = O(log(mn/ǫ)(log log(mn/ǫ)) 2 ), meaning:…”

“…We define the Kolmogorov distance between two integer valued random variables Z, Z ′ by d K (Z, Z ′ ) = max k∈Z (|Pr[Z ≤ k] − Pr[Z ′ ≤ k]|). Let X ′ ∈ [M ′ ] n be generated via the (M ′ , n)-fourier shape PRG of [GKM15] with error ǫ ′ (Theorem 1.1 [GKM15]). Observe…”

“…On top of this, the PRG of Section 5 [GKM15] first splits the coordinates [n] via a limited independence hash function into poly(log(1/ǫ)) buckets, and applies the algorithm described above on each. To do this second layer of bucketing and not need fresh randomness for each bucket, they use Nisan's PRG [Nis92] with a seed of length log(n) log log(n).…”