A Forward-Backward Weight Analysis of INW for Permutation Branching Programs

2026-07-20

AI Summary

This work addresses permutation branching programs of bounded width and proposes a novel program-structure-based forward-backward weight semi-norm analysis within the INW pseudorandom generator (PRG) framework, effectively controlling error propagation. The method achieves, for the first time, an optimal logarithmic dependence on program length \(n\) while simultaneously preserving an exponential improvement in the width \(w\). Specifically, the constructed \(\varepsilon\)-error PRG attains a seed length of \(O((\log w + \log(1/\varepsilon)) \cdot \log n)\), which is information-theoretically optimal in \(n\) and matches the current lower bound derived from spectral analysis.

Abstract

We construct an $\varepsilon$-error PRG for permutation read-once branching programs of length $n$ and width $w$ with seed length \[ O\left((\log w+\log(1/\varepsilon))\cdot \log n\right). \] This gives an exponential improvement in the dependence on $w$ compared with the constructions of De (CCC 2011) and Steinke (ECCC 2012). Compared with the work of Braverman, Rao, Raz, and Yehudayoff (FOCS 2010; SICOMP 2014), which applies more generally to regular branching programs and already achieves the optimal dependence on $w$, our result improves the dependence on the length $n$, attaining the optimal logarithmic dependence. The generator itself is the classical INW PRG of Impagliazzo, Nisan, and Wigderson (STOC 1994). We show that, for permutation branching programs, the INW generator can be instantiated with expanders whose degrees are polynomial in $w$ and $1/\varepsilon$ and, crucially, independent of $n$. To prove this, we analyze error propagation using program-dependent seminorms tailored to the branching program at hand. These seminorms build on the weight function introduced by Braverman et al. The key point is that, when measured in these adapted seminorms, the error does not accumulate throughout the recursion. Since our analysis relies only on the spectral expansion of the underlying expanders, our seed length tightly matches the recent lower bound for spectral analyses of the INW generator due to Hoza, Pyne, and Vadhan (Algorithmica 2024).

Problem

• pseudorandom generator
• permutation branching programs
• seed length
• read-once branching programs
• INW generator

Innovation

• Permutation Branching Programs
• Pseudorandom Generators
• INW Generator
• Spectral Expanders
• Program-Dependent Seminorms