Majority vs. Approximate Linear Sum and Average-Case Complexity Below NC¹

Lijie Chen, Zhenjian Lu, Xin Lyu, Igor C. Oliveira, Nikhil Bansal, Emanuela Merelli, James Worrell
We develop a general framework that characterizes strong average-case lower bounds against circuit classes 𝒞 contained in NC¹, such as AC⁰[⊕] and ACC⁰. We apply this framework to show: - Generic seed reduction: Pseudorandom generators (PRGs) against 𝒞 of seed length ≤ n -1 and error ε(n) = n^{-ω(1)} can be converted into PRGs of sub-polynomial seed length. - Hardness under natural distributions: If 𝖤 (deterministic exponential time) is average-case hard against 𝒞 under some distribution, then 𝖤
more » ... is average-case hard against 𝒞 under the uniform distribution. - Equivalence between worst-case and average-case hardness: Worst-case lower bounds against MAJ∘𝒞 for problems in 𝖤 are equivalent to strong average-case lower bounds against 𝒞. This can be seen as a certain converse to the Discriminator Lemma [Hajnal et al., JCSS'93]. These results were not known to hold for circuit classes that do not compute majority. Additionally, we prove that classical and recent approaches to worst-case lower bounds against ACC⁰ via communication lower bounds for NOF multi-party protocols [Håstad and Goldmann, CC'91; Razborov and Wigderson, IPL'93] and Torus polynomials degree lower bounds [Bhrushundi et al., ITCS'19] also imply strong average-case hardness against ACC⁰ under the uniform distribution. Crucial to these results is the use of non-black-box hardness amplification techniques and the interplay between Majority (MAJ) and Approximate Linear Sum (SUM̃) gates. Roughly speaking, while a MAJ gate outputs 1 when the sum of the m input bits is at least m/2, a SUM̃ gate computes a real-valued bounded weighted sum of the input bits and outputs 1 (resp. 0) if the sum is close to 1 (resp. close to 0), with the promise that one of the two cases always holds. As part of our framework, we explore ideas introduced in [Chen and Ren, STOC'20] to show that, for the purpose of proving lower bounds, a top layer MAJ gate is equivalent to a (weaker) SUM̃ gate. Motivated by this result, we extend the algorithmic method and establish stro [...]
doi:10.4230/lipics.icalp.2021.51 fatcat:mdtczu5akfh67fodxdvwe3w2bq