34 research outputs found
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Complexity Theory
Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness and randomness extraction. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes
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
Applications of Derandomization Theory in Coding
Randomized techniques play a fundamental role in theoretical computer science
and discrete mathematics, in particular for the design of efficient algorithms
and construction of combinatorial objects. The basic goal in derandomization
theory is to eliminate or reduce the need for randomness in such randomized
constructions. In this thesis, we explore some applications of the fundamental
notions in derandomization theory to problems outside the core of theoretical
computer science, and in particular, certain problems related to coding theory.
First, we consider the wiretap channel problem which involves a communication
system in which an intruder can eavesdrop a limited portion of the
transmissions, and construct efficient and information-theoretically optimal
communication protocols for this model. Then we consider the combinatorial
group testing problem. In this classical problem, one aims to determine a set
of defective items within a large population by asking a number of queries,
where each query reveals whether a defective item is present within a specified
group of items. We use randomness condensers to explicitly construct optimal,
or nearly optimal, group testing schemes for a setting where the query outcomes
can be highly unreliable, as well as the threshold model where a query returns
positive if the number of defectives pass a certain threshold. Finally, we
design ensembles of error-correcting codes that achieve the
information-theoretic capacity of a large class of communication channels, and
then use the obtained ensembles for construction of explicit capacity achieving
codes.
[This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi
Leakage-resilient coin tossing
Proceedings 25th International Symposium, DISC 2011, Rome, Italy, September 20-22, 2011.The ability to collectively toss a common coin among n parties
in the presence of faults is an important primitive in the arsenal of
randomized distributed protocols. In the case of dishonest majority, it
was shown to be impossible to achieve less than 1
r bias in O(r) rounds
(Cleve STOC â86). In the case of honest majority, in contrast, unconditionally
secure O(1)-round protocols for generating common unbiased
coins follow from general completeness theorems on multi-party secure
protocols in the secure channels model (e.g., BGW, CCD STOC â88).
However, in the O(1)-round protocols with honest majority, parties
generate and hold secret values which are assumed to be perfectly hidden
from malicious parties: an assumption which is crucial to proving the
resulting common coin is unbiased. This assumption unfortunately does
not seem to hold in practice, as attackers can launch side-channel attacks
on the local state of honest parties and leak information on their secrets.
In this work, we present an O(1)-round protocol for collectively generating
an unbiased common coin, in the presence of leakage on the local
state of the honest parties. We tolerate t †( 1
3
â )n computationallyunbounded
Byzantine faults and in addition a Ω(1)-fraction leakage on
each (honest) partyâs secret state. Our results hold in the memory leakage
model (of Akavia, Goldwasser, Vaikuntanathan â08) adapted to the
distributed setting.
Additional contributions of our work are the tools we introduce to
achieve the collective coin toss: a procedure for disjoint committee election,
and leakage-resilient verifiable secret sharing.National Defense Science and Engineering Graduate FellowshipNational Science Foundation (U.S.) (CCF-1018064
Computational Extractors with Negligible Error in the CRS Model
In recent years, there has been exciting progress on building two-source extractors for sources with low min-entropy. Unfortunately, all known explicit constructions of two-source extractors in the low entropy regime suffer from non-negligible error, and building such extractors with negligible error remains an open problem. We investigate this problem in the computational setting, and obtain the following results.
We construct an explicit 2-source extractor, and even an explicit non-malleable extractor, with negligible error, for sources with low min-entropy, under computational assumptions in the Common Random String (CRS) model. More specifically, we assume that a CRS is generated once and for all, and allow the min-entropy sources to depend on the CRS. We obtain our
constructions by using the following transformations.
1. Building on the technique of [BHK11], we show a general transformation for converting any computational 2-source extractor (in the CRS model) into a computational non-malleable extractor (in the CRS model), for sources with similar min-entropy. We emphasize that the resulting computational non-malleable extractor is resilient to arbitrarily many tampering attacks (a property that is impossible to achieve information theoretically). This may be of independent interest. This transformation uses cryptography, and relies on the sub-exponential hardness of the Decisional Diffie Hellman (DDH) assumption.
2. Next, using the blueprint of [BACDLT17], we give a transformation converting our computational non-malleable seeded extractor (in the CRS model) into a computational 2-source extractor for sources with low min-entropy (in the CRS model).
Our 2-source extractor works for unbalanced sources: specifically, we require one of the sources to be larger than a specific polynomial in the other. This transformation does not incur any additional assumptions. Our analysis makes a novel use of the leakage lemma of Gentry and Wichs [GW11]
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Distributed computing and cryptography with general weak random sources
The use of randomness in computer science is ubiquitous. Randomized protocols have turned out to be much more efficient than their deterministic counterparts. In addition, many problems in distributed computing and cryptography are impossible to solve without randomness. However, these applications typically require uniform random bits, while in practice almost all natural random phenomena are biased. Moreover, even originally uniform random bits can be damaged if an adversary learns some partial information about these bits. In this thesis, we study how to run randomized protocols in distributed computing and cryptography with imperfect randomness. We use the most general model for imperfect randomness where the weak random source is only required to have a certain amount of min-entropy. One important tool here is the randomness extractor. A randomness extractor is a function that takes as input one or more weak random sources, and outputs a distribution that is close to uniform in statistical distance. Randomness extractors are interesting in their own right and are closely related to many other problems in computer science. Giving efficient constructions of randomness extractors with optimal parameters is one of the major open problems in the area of pseudorandomness. We construct network extractor protocols that extract private random bits for parties in a communication network, assuming that they each start with an independent weak random source, and some parties are corrupted by an adversary who sees all communications in the network. These protocols imply fault-tolerant distributed computing protocols and secure multi-party computation protocols where only imperfect randomness is available. The probabilistic method shows that there exists an extractor for two independent sources with logarithmic min-entropy, while known constructions are far from achieving these parameters. In this thesis we construct extractors for two independent sources with any linear min-entropy, based on a computational assumption. We also construct the best known extractors for three independent sources and affine sources. Finally we study the problem of privacy amplification. In this model, two parties share a private weak random source and they wish to agree on a private uniform random string through communications in a channel controlled by an adversary, who has unlimited computational power and can change the messages in arbitrary ways. All previous results assume that the two parties have local uniform random bits. We show that this problem can be solved even if the two parties only have local weak random sources. We also improve previous results in various aspects by constructing the first explicit non-malleable extractor and giving protocols based on this extractor.Computer Science
Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More
A long line of work in the past two decades or so established close
connections between several different pseudorandom objects and applications.
These connections essentially show that an asymptotically optimal construction
of one central object will lead to asymptotically optimal solutions to all the
others. However, despite considerable effort, previous works can get close but
still lack one final step to achieve truly asymptotically optimal
constructions.
In this paper we provide the last missing link, thus simultaneously achieving
explicit, asymptotically optimal constructions and solutions for various well
studied extractors and applications, that have been the subjects of long lines
of research. Our results include:
Asymptotically optimal seeded non-malleable extractors, which in turn give
two source extractors for asymptotically optimal min-entropy of ,
explicit constructions of -Ramsey graphs on vertices with , and truly optimal privacy amplification protocols with an active adversary.
Two source non-malleable extractors and affine non-malleable extractors for
some linear min-entropy with exponentially small error, which in turn give the
first explicit construction of non-malleable codes against -split state
tampering and affine tampering with constant rate and \emph{exponentially}
small error.
Explicit extractors for affine sources, sumset sources, interleaved sources,
and small space sources that achieve asymptotically optimal min-entropy of
or (for space sources).
An explicit function that requires strongly linear read once branching
programs of size , which is optimal up to the constant in
. Previously, even for standard read once branching programs, the
best known size lower bound for an explicit function is .Comment: Fixed some minor error
Non-malleable coding against bit-wise and split-state tampering
Non-malleable coding, introduced by Dziembowski et al. (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Non-malleable coding is possible against any class of adversaries of bounded size. In particular, Dziembowski et al. show that such codes exist and may achieve positive rates for any class of tampering functions of size at most (Formula presented.), for any constant (Formula presented.). However, this result is existential and has thus attracted a great deal of subsequent research on explicit constructions of non-malleable codes against natural classes of adversaries. In this work, we consider constructions of coding schemes against two well-studied classes of tampering functions; namely, bit-wise tampering functions (where the adversary tampers each bit of the encoding independently) and the much more general class of split-state adversaries (where two independent adversaries arbitrarily tamper each half of the encoded sequence). We obtain the following results for these models. (1) For bit-tampering adversaries, we obtain explicit and efficiently encodable and decodable non-malleable codes of length n achieving rate (Formula presented.) and error (also known as âexact securityâ) (Formula presented.). Alternatively, it is possible to improve the error to (Formula presented.) at the cost of making the construction Monte Carlo with success probability (Formula presented.) (while still allowing a compact description of the code). Previously, the best known construction of bit-tampering coding schemes was due to Dziembowski et al. (ICS 2010), which is a Monte Carlo construction achieving rate close to .1887. (2) We initiate the study of seedless non-malleable extractors as a natural variation of the notion of non-malleable extractors introduced by Dodis and Wichs (STOC 2009). We show that construction of non-malleable codes for the split-state model reduces to construction of non-malleable two-source extractors. We prove a general result on existence of seedless non-malleable extractors, which implies that codes obtained from our reduction can achieve rates arbitrarily close to 1 / 5 and exponentially small error. In a separate recent work, the authors show that the optimal rate in this model is 1 / 2. Currently, the best known explicit construction of split-state coding schemes is due to Aggarwal, Dodis and Lovett (ECCC TR13-081) which only achieves vanishing (polynomially small) rate
Privacy Amplification with Tamperable Memory via Non-malleable Two-source Extractors
We extend the classical problem of privacy amplification to a setting where the active adversary, Eve, is also allowed to fully corrupt the internal memory (which includes the shared randomness, and local randomness tape) of one of the honest parties, Alice and Bob, before the execution of the protocol. We require that either one of Alice or Bob detects tampering, or they agree on a shared key that is indistinguishable from the uniform distribution to Eve. We obtain the following results:
(1) We give a privacy amplification protocol via low-error non-malleable two-source extractors with one source having low min-entropy. In particular, this implies the existence of such (non-efficient) protocols;
(2) We show that even slight improvements to the state-of-the-art explicit non-malleable two-source extractors would lead to explicit low-error, low min-entropy two-source extractors, thereby resolving a long-standing open question. This suggests that obtaining (information-theoretically secure) explicit non-malleable two-source extractors for (1) might be hard;
(3) We present explicit constructions of low-error, low min-entropy non-malleable two-source extractors in the CRS model of (Garg, Kalai, Khurana, Eurocrypt 2020), assuming either the quasi-polynomial hardness of DDH or the existence of nearly-optimal collision-resistant hash functions;
(4) We instantiate our privacy amplification protocol with the above mentioned non-malleable two-source extractors in the CRS model, leading to explicit, computationally-secure protocols. This is not immediate from (1) because in the computational setting we need to make sure that, in particular, all randomness sources remain samplable throughout the
proof. This requires upgrading the assumption of quasi-polynomial hardness of DDH to sub-exponential hardness of DDH.
We emphasize that each of the first three results can be read independently