TY - GEN
T1 - On beating the hybrid argument
AU - Fefferman, Bill
AU - Shaltiel, Ronen
AU - Umans, Christopher
AU - Viola, Emanuele
PY - 2012
Y1 - 2012
N2 - 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.
AB - 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.
KW - branching program
KW - constant depth circuits
KW - hybrid argument
KW - pseudorandomness
KW - quantum computing
KW - small space
UR - http://www.scopus.com/inward/record.url?scp=84856501579&partnerID=8YFLogxK
U2 - 10.1145/2090236.2090273
DO - 10.1145/2090236.2090273
M3 - Conference contribution
AN - SCOPUS:84856501579
SN - 9781450311151
T3 - ITCS 2012 - Innovations in Theoretical Computer Science Conference
SP - 468
EP - 483
BT - ITCS 2012 - Innovations in Theoretical Computer Science Conference
T2 - 3rd Conference on Innovations in Theoretical Computer Science, ITCS 2012
Y2 - 8 January 2012 through 10 January 2012
ER -