Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs
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In a recent work, Gryaznov, Pudlák, and Talebanfard (CCC' 22) introduced a stronger version of affine extractors known as directional affine extractors, together with a generalization of 𝖱𝖮𝖡𝖯s where each node can make linear queries, and showed that the former implies strong lower bound for a certain type of the latter known as strongly read-once linear branching programs (𝖲𝖱𝖮𝖫𝖡𝖯s). Their main result gives explicit constructions of directional affine extractors for entropy k > 2n/3, which implies average-case complexity 2^n/3-o(n) against 𝖲𝖱𝖮𝖫𝖡𝖯s with exponentially small correlation. A follow-up work by Chattopadhyay and Liao (ECCC' 22) improves the hardness to 2^n-o(n) at the price of increasing the correlation to polynomially large. This paper provides a much more in-depth study of directional affine extractors, 𝖲𝖱𝖮𝖫𝖡𝖯s, and 𝖱𝖮𝖡𝖯s. Our main results include: A formal separation between 𝖲𝖱𝖮𝖫𝖡𝖯 and 𝖱𝖮𝖡𝖯, showing that 𝖲𝖱𝖮𝖫𝖡𝖯s can be exponentially more powerful than 𝖱𝖮𝖡𝖯s. An explicit construction of directional affine extractors with k=o(n) and exponentially small error, which gives average-case complexity 2^n-o(n) against 𝖲𝖱𝖮𝖫𝖡𝖯s with exponentially small correlation, thus answering the two open questions raised in previous works. An explicit function in 𝖠𝖢^0 that gives average-case complexity 2^(1-δ)n against 𝖱𝖮𝖡𝖯s with negligible correlation, for any constant δ>0. Previously, the best size lower bound for any function in 𝖠𝖢^0 against 𝖱𝖮𝖡𝖯s is only 2^Ω(√(n)). One of the key ingredients in our constructions is a new linear somewhere condenser for affine sources.
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