Limitations of the Impagliazzo–Nisan–Wigderson Pseudorandom Generator Against Permutation Branching Programs

The classic Impagliazzo–Nisan–Wigderson (INW) pseudorandom generator (PRG) (STOC ‘94) for space-bounded computation uses a seed of length \(O(\log n \cdot \log (nw/\varepsilon )+\log d)\) to fool ordered branching programs of length n, width w, and alphabet size d to within error \(\varepsilon \). A series of works have shown that the analysis of the INW generator can be improved for the class of permutation branching programs or the more general regular branching programs, improving the \(O(\log ^2 n)\) dependence on the length n to \(O(\log n)\) or \({\tilde{O}}(\log n)\). However, when also considering the dependence on the other parameters, these analyses still fall short of the optimal PRG seed length \(O(\log (nwd/\varepsilon ))\). In this paper, we prove that any “spectral analysis” of the INW generator requires seed length

$$\begin{aligned} \Omega \left( \log n\cdot \log \log \left( \min \{n,d\}\right) +\log n\cdot \log \left( w/\varepsilon \right) +\log d\right) \end{aligned}$$

to fool ordered permutation branching programs of length n, width w, and alphabet size d to within error \(\varepsilon \). By “spectral analysis” we mean an analysis of the INW generator that relies only on the spectral expansion of the graphs used to construct the generator; this encompasses all prior analyses of the INW generator. Our lower bound matches the upper bound of Braverman–Rao–Raz–Yehudayoff (FOCS 2010, SICOMP 2014) for regular branching programs of alphabet size \(d=2\) except for a gap between their \(O\left( \log n \cdot \log \log n\right) \) term and our \(\Omega \left( \log n \cdot \log \log \min \{n,d\}\right) \) term. It also matches the upper bounds of Koucký–Nimbhorkar–Pudlák (STOC 2011), De (CCC 2011), and Steinke (ECCC 2012) for constant-width (\(w=O(1)\)) permutation branching programs of alphabet size \(d=2\) to within a constant factor. To fool permutation branching programs in the measure of spectral norm, we prove that any spectral analysis of the INW generator requires a seed of length \(\Omega \left( \log n\cdot \log \log n+\log n\cdot \log (1/\varepsilon )\right) \) when the width is at least polynomial in n (\(w=n^{\Omega (1)}\)), matching the recent upper bound of Hoza–Pyne–Vadhan (ITCS 2021) to within a constant factor.