Bounded Independence Plus Noise Fools Products

Let D be a b-wise independent distribution over {0,1}^m. Let E be the "noise" distribution over {0,1}^m where the bits are independent and each bit is 1 with probability eta/2. We study which tests f: {0,1}^m -> [-1,1] are epsilon-fooled by D+E, i.e., |E[f(D+E)] - E[f(U)]| <= epsilon where U is the uniform distribution. We show that D+E epsilon-fools product tests f: ({0,1}^n)^k -> [-1,1] given by the product of k bounded functions on disjoint n-bit inputs with error epsilon = k(1-eta)^{Omega(b^2/m)}, where m = nk and b >= n. This bound is tight when b = Omega(m) and eta >= (log k)/m. For b >= m^{2/3} log m and any constant eta the distribution D+E also 0.1-fools log-space algorithms. We develop two applications of this type of results. First, we prove communication lower bounds for decoding noisy codewords of length m split among k parties. For Reed-Solomon codes of dimension m/k where k = O(1), communication Omega(eta m) - O(log m) is required to decode one message symbol from a codeword with eta m errors, and communication O(eta m log m) suffices. Second, we obtain pseudorandom generators. We can epsilon-fool product tests f: ({0,1}^n)^k -> [-1,1] under any permutation of the bits with seed lengths 2n + O~(k^2 log(1/epsilon)) and O(n) + O~(sqrt{nk log 1/epsilon}). Previous generators have seed lengths >= nk/2 or >= n sqrt{n k}. For the special case where the k bounded functions have range {0,1} the previous generators have seed length >= (n+log k)log(1/epsilon)

Pseudorandom Bits for Oblivious Branching Programs

We construct a pseudorandom generator that fools known-order read-k oblivious branching programs and, more generally, any linear length oblivious branching program. For polynomial width branching programs, the seed lengths in our constructions are O(n^(1−1/2^(k−1))) (for the read-k case) and O(n/log log n) (for the linear length case). Previously, the best construction for these models required seed length (1 − Ω(1))n

Pseudorandom Generators for Width-3 Branching Programs

We construct pseudorandom generators of seed length $\tilde{O}(\log(n)\cdot \log(1/\epsilon))$ that $\epsilon$-fool ordered read-once branching programs (ROBPs) of width $3$ and length $n$. For unordered ROBPs, we construct pseudorandom generators with seed length $\tilde{O}(\log(n) \cdot \mathrm{poly}(1/\epsilon))$. This is the first improvement for pseudorandom generators fooling width $3$ ROBPs since the work of Nisan [Combinatorica, 1992]. Our constructions are based on the `iterated milder restrictions' approach of Gopalan et al. [FOCS, 2012] (which further extends the Ajtai-Wigderson framework [FOCS, 1985]), combined with the INW-generator [STOC, 1994] at the last step (as analyzed by Braverman et al. [SICOMP, 2014]). For the unordered case, we combine iterated milder restrictions with the generator of Chattopadhyay et al. [CCC, 2018]. Two conceptual ideas that play an important role in our analysis are: (1) A relabeling technique allowing us to analyze a relabeled version of the given branching program, which turns out to be much easier. (2) Treating the number of colliding layers in a branching program as a progress measure and showing that it reduces significantly under pseudorandom restrictions. In addition, we achieve nearly optimal seed-length $\tilde{O}(\log(n/\epsilon))$ for the classes of: (1) read-once polynomials on $n$ variables, (2) locally-monotone ROBPs of length $n$ and width $3$ (generalizing read-once CNFs and DNFs), and (3) constant-width ROBPs of length $n$ having a layer of width $2$ in every consecutive $\mathrm{poly}\log(n)$ layers.Comment: 51 page