24 research outputs found
Non-Malleable Extractors and Codes, with their Many Tampered Extensions
Randomness extractors and error correcting codes are fundamental objects in
computer science. Recently, there have been several natural generalizations of
these objects, in the context and study of tamper resilient cryptography. These
are seeded non-malleable extractors, introduced in [DW09]; seedless
non-malleable extractors, introduced in [CG14b]; and non-malleable codes,
introduced in [DPW10].
However, explicit constructions of non-malleable extractors appear to be
hard, and the known constructions are far behind their non-tampered
counterparts.
In this paper we make progress towards solving the above problems. Our
contributions are as follows.
(1) We construct an explicit seeded non-malleable extractor for min-entropy
. This dramatically improves all previous results and gives a
simpler 2-round privacy amplification protocol with optimal entropy loss,
matching the best known result in [Li15b].
(2) We construct the first explicit non-malleable two-source extractor for
min-entropy , with output size and
error .
(3) We initiate the study of two natural generalizations of seedless
non-malleable extractors and non-malleable codes, where the sources or the
codeword may be tampered many times. We construct the first explicit
non-malleable two-source extractor with tampering degree up to
, which works for min-entropy , with
output size and error . We show that we can
efficiently sample uniformly from any pre-image. By the connection in [CG14b],
we also obtain the first explicit non-malleable codes with tampering degree
up to , relative rate , and error
.Comment: 50 pages; see paper for full abstrac
Optimal Error Pseudodistributions for Read-Once Branching Programs
In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom
generator for length and width read-once branching programs with seed
length and error
. It remains a central question to reduce the seed length to
, which would prove that .
However, there has been no improvement on Nisan's construction for the case
, which is most relevant to space-bounded derandomization.
Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced
the notion of a pseudorandom pseudo-distribution (PRPD) and gave an explicit
construction of a PRPD with seed length . A PRPD is a relaxation of a pseudorandom
generator, which suffices for derandomizing and also implies a
hitting set. Unfortunately, their construction is quite involved and
complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler
hitting set generator with seed length , but their techniques are restricted to hitting
sets.
In this work, we construct a PRPD with seed length This improves upon the
construction in [BCG18] by a factor, and is
optimal in the small error regime. In addition, we believe our construction and
analysis to be simpler than the work of Braverman, Cohen and Garg
Recursive Error Reduction for Regular Branching Programs
In a recent work, Chen, Hoza, Lyu, Tal and Wu (FOCS 2023) showed an improved
error reduction framework for the derandomization of regular read-once
branching programs (ROBPs). Their result is based on a clever modification to
the inverse Laplacian perspective of space-bounded derandomization, which was
originally introduced by Ahmadinejad, Kelner, Murtagh, Peebles, Sidford and
Vadhan (FOCS 2020).
In this work, we give an alternative error reduction framework for regular
ROBPs. Our new framework is based on a binary recursive formula from the work
of Chattopadhyay and Liao (CCC 2020), that they used to construct weighted
pseudorandom generators (WPRGs) for general ROBPs.
Based on our new error reduction framework, we give alternative proofs to the
following results for regular ROBPs of length and width , both of which
were proved in the work of Chen et al. using their error reduction:
There is a WPRG with error that has seed length
There is a (non-black-box) deterministic algorithm which estimates
the expectation of any such program within error with space
complexity (This was first
proved in the work of Ahmadinejad et al., but the proof by Chen et al. is
simpler.)
Because of the binary recursive nature of our new framework, both of our
proofs are based on a straightforward induction that is arguably simpler than
the Laplacian-based proof in the work of Chen et al
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Explicit two-source extractors and more
In this thesis we study the problem of extracting almost truly random bits from imperfect sources of randomness. This is motivated by the wide use of randomness in computer science, and the fact that most accessible sources of randomness generate correlated bits, and at best contain some amount of entropy. We follow Chor and Goldreich [CG88] and Zuckerman [Z90], and model weak sources using min-entropy, where an (n,k)-source X is a distribution on n bits and takes any string x with probability at most 2^-k. It is known that it is impossible to extract random bits from a single (n,k)-source, and Chor and Goldreich [CG88] raised the question of extracting randomness from two such independent (n,k)-sources. Existentially, such 2-source randomness extractors exist for min-entropy k >=log n + O(1), but the best known construction prior to work in this thesis requires min-entropy k >=0.499 n [B2]. One of the main contributions of this thesis is an explicit 2-source extractor for min-entropy log^C n, for some constant C. Other results in this thesis include improved ways of extracting random bits from various other sources of randomness, as well as stronger notions of randomness extraction. Our results have applications in privacy amplification [BBR88,Mau92,BBCM95], which is a classical problem in information cryptography, and give protocols that achieve almost optimal parameters. Other applications include explicit constructions of non-malleable codes, which is a relaxation of the notion of error-detection codes and have applications in tamper-resilient cryptography [DPW10].Computer Science