Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs

In a recent work, Gryaznov, Pudl\'{a}k, and Talebanfard (CCC' 22) introduced a stronger version of affine extractors known as directional affine extractors, together with a generalization of $\mathsf{ROBP}$s 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 ($\mathsf{SROLBP}$s). 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 $\mathsf{SROLBP}$s 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, $\mathsf{SROLBP}$s, and $\mathsf{ROBP}$s. Our main results include: A formal separation between $\mathsf{SROLBP}$ and $\mathsf{ROBP}$, showing that $\mathsf{SROLBP}$s can be exponentially more powerful than $\mathsf{ROBP}$s. 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 $\mathsf{SROLBP}$s with exponentially small correlation, thus answering the two open questions raised in previous works. An explicit function in $\mathsf{AC}^0$ that gives average-case complexity $2^{(1-\delta)n}$ against $\mathsf{ROBP}$s with negligible correlation, for any constant $\delta>0$. Previously, the best size lower bound for any function in $\mathsf{AC}^0$ against $\mathsf{ROBP}$s is only $2^{\Omega(\sqrt{n})}$. One of the key ingredients in our constructions is a new linear somewhere condenser for affine sources.