On beating the hybrid argument

Bill Fefferman, Ronen Shaltiel, Christopher Umans, Emanuele Viola

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


The hybrid argument allows one to relate the distinguishability of a distribution (from uniform) to the predictability of individual bits given a prefix. The argument incurs a loss of a factor k equal to the bit-length of the distributions: ε-distinguishability implies ε/k-predictability. This paper studies the consequences of avoiding this loss - what we call "beating the hybrid argument" - and develops new proof techniques that circumvent the loss in certain natural settings. Specifically, we obtain the following results: 1. We give an instantiation of the Nisan-Wigderson generator (JCSS '94) that can be broken by quantum computers, and that is o(1)-unpredictable against AC 0. We conjecture that this generator indeed fools AC 0. Our conjecture implies the existence of an oracle relative to which BQP is not in the PH, a longstanding open problem. 2. We show that the "INW" generator by Impagliazzo, Nisan, and Wigderson (STOC '94) with seed length O(log n log log n) produces a distribution that is 1/log n-unpredictable against poly-logarithmic width (general) read-once oblivious branching programs. Obtaining such generators where the output is indistinguishable from uniform is a longstanding open problem. 3. We identify a property of functions f, "resamplability," that allows us to beat the hybrid argument when arguing indistinguishability of (Equation Presented) from uniform. This gives new pseudorandom generators for classes such as AC 0[p] with a stretch that, despite being sub-linear, is the largest known. We view this as a first step towards beating the hybrid argument in the analysis of the Nisan-Wigderson generator (which applies (Equation Presented) on correlated x 1,...,x k) and proving the conjecture in the first item.

Original languageEnglish
Title of host publicationITCS 2012 - Innovations in Theoretical Computer Science Conference
Number of pages16
StatePublished - 2012
Event3rd Conference on Innovations in Theoretical Computer Science, ITCS 2012 - Cambridge, MA, United States
Duration: 8 Jan 201210 Jan 2012

Publication series

NameITCS 2012 - Innovations in Theoretical Computer Science Conference


Conference3rd Conference on Innovations in Theoretical Computer Science, ITCS 2012
Country/TerritoryUnited States
CityCambridge, MA


  • branching program
  • constant depth circuits
  • hybrid argument
  • pseudorandomness
  • quantum computing
  • small space

ASJC Scopus subject areas

  • Computational Theory and Mathematics


Dive into the research topics of 'On beating the hybrid argument'. Together they form a unique fingerprint.

Cite this