Electronic Colloquium on Computational Complexity
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TR17-027 | 16th February 2017 22:30

A reduction from efficient non-malleable extractors to low-error two-source extractors with arbitrary constant rate


Authors: Avraham Ben-Aroya, Eshan Chattopadhyay, Dean Doron, Xin Li, Amnon Ta-Shma
Publication: 17th February 2017 16:20
Downloads: 60


We show a reduction from the existence of explicit t-non-malleable
extractors with a small seed length, to the construction of explicit
two-source extractors with small error for sources with arbitrarily
small constant rate. Previously, such a reduction was known either
when one source had entropy rate above half [Raz05] or for
general entropy rates but only for large error [CZ16].

As in previous reductions we start with the Chattopadhyay and
Zuckerman approach [CZ16], where an extractor is applied on one
source to create a table, and the second source is used to sample a
sub-table. In previous work, a resilient function was applied on the
sub-table and the use of resilient functions immediately implied
large error. In this work we replace the resilient function with the
parity function (that is not resilient). We prove correctness by
showing that doing the sampling properly, the sample size can be
made so small that it is smaller then the non-malleability parameter
t of the first extractor, which is enough for the correctness.

The parameters we require from the non-malleable construction hold
(quite comfortably) in a non-explicit construction, but currently it
is not known how to explicitly construct such graphs. As a result we
do not give an unconditional construction of an explicit low-error
two-source extractor. However, the reduction shows a new connection
between non-malleable and two-source extractors, and also shows
resilient functions do not play a crucial role in the two-source
construction framework suggested in [CZ16]. Furthermore, the
reduction highlights a barrier in constructing non-malleable
extractors, and reveals its importance. We hope this work would lead
to further developments in explicit constructions of both
non-malleable and two-source extractors.

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