Theory of Computing
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Title : On Beating the Hybrid Argument
Authors : Bill Fefferman, Ronen Shaltiel, Christopher Umans, and Emanuele Viola
Volume : 9
Number : 26
Pages : 809-843
URL : https://theoryofcomputing.org/articles/v009a026
Abstract
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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: $\epsilon$-distinguishability
implies $\epsilon/k$-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. Our main results are:
* 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.
* 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.
(This was also observed by other researchers.) Obtaining such
generators where the output is indistinguishable from uniform is a
longstanding open problem.
* We identify a property of functions $f$, "resamplability," that
allows us to beat the hybrid argument when arguing
indistinguishability of \[G^{\otimes k}_f(x_1,\ldots,x_k) =
(x_1,f(x_1),x_2,f(x_2),\ldots,x_k,f(x_k))\] 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
$G^{\otimes k}_f$ on correlated $x_1,\ldots,x_k$) and proving the
conjecture in the first item.
An extended abstract of this paper appeared in the Proceedings of the
3rd Innovations in Theoretical Computer Science (ITCS) 2012.