Online Condensing of Unpredictable Sources via Random Walks

A natural model of a source of randomness consists of a long stream of symbols X = X1 ◦... ◦ Xt, with some guarantee on the entropy of Xi conditioned on the outcome of the prefix x₁, ..., x_i−1. We study unpredictable sources, a generalization of the almost Chor-Goldreich (CG) sources considered in [9]. In an unpredictable source X, for a typical draw of x ∼ X, for most i-s, the element xi has a low probability of occurring given x1, ..., xi−1. Such a model relaxes the often unrealistic assumption of a CG source that for every i, and every x1, ..., xi−1, the next symbol Xi has sufficiently large entropy. Unpredictable sources subsume all previously considered notions of almost CG sources, including notions that [9] failed to analyze, and including those that are equivalent to general sources with high min entropy. For a lossless expander G = (V, E) with m = log |V |, we consider a random walk V0, V1, ..., Vt on G using unpredictable instructions that have sufficient entropy with respect to m. Our main theorem is that for almost all the steps t/2 ≤ i ≤ t in the walk, the vertex Vi is close to a distribution with min-entropy at least m − O(1). As a result, we obtain seeded online condensers with constant entropy gap, and seedless (deterministic) condensers outputting a constant fraction of the entropy. In particular, our condensers run in space comparable to the output entropy, as opposed to the size of the stream, and even when the length t of the stream is not known ahead of time. As another corollary, we obtain a new extractor based on expander random walks handling lower entropy than the classic expander based construction relying on spectral techniques [11]. As our main technical tool, we provide a novel analysis covering a key case of adversarial random walks on lossless expanders that [9] fails to address. As part of the analysis, we provide a “chain rule for vertex probabilities”. The standard chain rule states that for every x ∼ X and i, Pr(x1, ..., xi) = Pr[Xi = xi|X_[1,i−_1] = x1, ..., xi−1]·Pr(x1, ..., xi−1). If W(x1, ..., xi) is the vertex reached using x1, ..., xi, then the chain rule for vertex probabilities essentially states that the same phenomena occurs for a typical x: Pr[Vi = W(x1, ..., xi)] ≲ Pr[Xi = xi|X_[1,i−_1] = x1, ..., xi−1] · Pr[Vi−1 = W(x1, ..., xi−1)], where Vi is the vertex distribution of the random walk at step i using X.