Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs
Xin Li, Yan Zhong
TL;DR
This work advances the theory of randomness extractors by introducing explicit directional affine extractors that operate for sublinear to linear entropy with exponentially small error, delivering near-optimal average-case hardness for strongly read-once linear branching programs. A central technical contribution is a linear somewhere condenser based on dimension expanders, which enables strengthening entropy guarantees and simplifying the construction of extractors across affine and general weak sources via the Polynomial Freiman-Ruzsa framework. The authors further develop AC$^0$-computable extractors and correlation breakers to obtain explicit average-case hardness against ROBP with negligible correlation, significantly improving prior bounds and addressing open questions. Together, these results illuminate new connections between affine source structure, algebraic pseudorandomness, and circuit/branching-program lower bounds, with implications for complexity theory and pseudorandomness in restricted models.
Abstract
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 $\mathsf{ROBP}$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. In this paper we show: 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 $2^{(1-δ)n}$ against $\mathsf{ROBP}$s with negligible correlation, for any constant $δ>0$. Previously, no such average-case hardness is known, and the best size lower bound for any function in $\mathsf{AC}^0$ against $\mathsf{ROBP}$s is $2^{Ω(n)}$. One of the key ingredients in our constructions is a new linear somewhere condenser for affine sources, which is based on dimension expanders. The condenser also leads to an unconditional improvement of the entropy requirement of explicit affine extractors with negligible error. We further show that the condenser also works for general weak random sources, under the Polynomial Freiman-Ruzsa Theorem in $\mathsf{F}_2^n$.