# Time-Space Tradeoffs for Distinguishing Distributions and Applications to Security of Goldreich's PRG

In this work, we establish lower-bounds against memory bounded algorithms for distinguishing between natural pairs of related distributions from samples that arrive in a streaming setting. In our first result, we show that any algorithm that distinguishes between uniform distribution on {0,1}^n and uniform distribution on an n/2-dimensional linear subspace of {0,1}^n with non-negligible advantage needs 2^Ω(n) samples or Ω(n^2) memory. Our second result applies to distinguishing outputs of Goldreich's local pseudorandom generator from the uniform distribution on the output domain. Specifically, Goldreich's pseudorandom generator G fixes a predicate P:{0,1}^k →{0,1} and a collection of subsets S_1, S_2, ..., S_m ⊆ [n] of size k. For any seed x ∈{0,1}^n, it outputs P(x_S_1), P(x_S_2), ..., P(x_S_m) where x_S_i is the projection of x to the coordinates in S_i. We prove that whenever P is t-resilient (all non-zero Fourier coefficients of (-1)^P are of degree t or higher), then no algorithm, with <n^ϵ memory, can distinguish the output of G from the uniform distribution on {0,1}^m with a large inverse polynomial advantage, for stretch m <(n/t)^(1-ϵ)/36· t (barring some restrictions on k). The lower bound holds in the streaming model where at each time step i, S_i⊆ [n] is a randomly chosen (ordered) subset of size k and the distinguisher sees either P(x_S_i) or a uniformly random bit along with S_i. Our proof builds on the recently developed machinery for proving time-space trade-offs (Raz 2016 and follow-ups) for search/learning problems.

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