Bounded Independence Plus Noise Fools Products

Abstract

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 η/2. We study which tests f : {0, 1} m → [−1, 1] are ε-fooled by D + E, i.e., | E[f (D + E)] − E[f (U)]| ≤ ε where U is the uniform distribution. We show that D + E ε-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 ε = k(1 − η) Ω(b 2 /m) , where m = nk and b ≥ n. This bound is tight when b = Ω(m) and η ≥ (log k)/m. For b ≥ m 2/3 log m and any constant η 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 Ω(ηm) − O(log m) is required to decode one message symbol from a codeword with ηm errors, and communication O(ηm log m) suffices. Second, we obtain pseudorandom generators. We can ε-fool product tests f : ({0, 1} n) k → [−1, 1] under any permutation of the bits with seed lengths 2n +Õ(k 2 log(1/ε)) and O(n) + O(nk log 1/ε). Previous generators have seed lengths ≥ nk/2 or ≥ n √ nk. For the special case where the k bounded functions have range {0, 1} the previous generators have seed length ≥ (n + log k) log(1/ε).

Pseudorandom Bits for Oblivious Branching Programs

“…In their construction, the seed length is (1 − Ω(1))n, whereas our seed length is sublinear in n. However, an advantage of their construction is that it works even when the reading order of the bits is unknown, while we require it to be known in advance. Haramaty, Lee and Viola obtained some improved bounds for the related model of product tests [HLV17].…”

Untitled

“…Handling arbitrary order is significantly more challenging, because the classical space-bounded generators [31,22] only work in fixed order [37,5]. Our previous work with Haramaty [20] gave the first generators for this class, but its dependence on k is poor: the seed length is always ≥ √ k. In this paper we improve the dependence on k exponentially, though the results in [20] remain unsurpassed when k is very small, e. g., k = O(1). We actually obtain two incomparable generators.…”

Pseudorandom generators for width-3 branching programs

“…PRGs for product-functions were constructed in previous work, however none achieve the parameters we need. Haramaty, Lee and Viola [HLV17] and Lee and Viola [LV17] constructed PRGs with seed length O(b + mb log(1/ε)) and O((b + log(m/ε)) · log(1/ε)) respectively for such functions. While the latter is nearly optimal for constant ε, we require ε to be smaller than 1/m, since the reduction in the previous section from [BDVY13] incurs a multiplicative factor of m on the error.…”