Online linear extractors for independent sources

In this work, we characterize linear online extractors. In other words, given a matrix A ∈ Fⁿ₂^×n, we study the convergence of the iterated process S ← AS ⊕ X, where X ∼ D is repeatedly sampled independently from some fixed (but unknown) distribution D with (min)-entropy k. Here, we think of S ∈ {0, 1}ⁿ as the state of an online extractor, and X ∈ {0, 1}ⁿ as its input. As our main result, we show that the state S converges to the uniform distribution for all input distributions D with entropy k > 0 if and only if the matrix A has no non-trivial invariant subspace (i.e., a non-zero subspace V ⊆ Fⁿ₂ such that AV ⊆ V ). In other words, a matrix A yields a linear online extractor if and only if A has no non-trivial invariant subspace. For example, the linear transformation corresponding to multiplication by a generator of the field F2n yields a good linear online extractor. Furthermore, for any such matrix convergence takes at most O^e(n²(k + 1)/k²) steps. We also study the more general notion of condensing - that is, we ask when this process converges to a distribution with entropy at least ℓ, when the input distribution has entropy at least k. (Extractors corresponding to the special case when ℓ = n.) We show that a matrix gives a good condenser if there are relatively few vectors w ∈ Fⁿ₂ such that w, A^T w,..., (A^T )ⁿ−^kw are linearly dependent. As an application, we show that the very simple cyclic rotation transformation A(x1,..., xn) = (xn, x1,..., xn−1) condenses to ℓ = n− 1 bits for any k > 1 if n is a prime satisfying a certain simple number-theoretic condition. Our proofs are Fourier-analytic and rely on a novel lemma, which gives a tight bound on the product of certain Fourier coefficients of any entropic distribution.