373 research outputs found
Probabilistic Proof Systems
Various types of probabilistic proof systems have played a central role in the development of computer science in the last decade. In this exposition, we concentrate on three such proof systems -- interactive proofs, zero-knowledge proofs, and probabilistic checkable proofs -- stressing the essential role of randomness in each of them. This exposition is an expanded version of a survey written for the proceedings of the International Congress of Mathematicians (ICM94) held in Zurich in 1994. It is hope that this exposition may be accessible to a broad audience of computer scientists and mathematians
The Random Oracle Methodology, Revisited
We take a critical look at the relationship between the security of
cryptographic schemes in the Random Oracle Model, and the security of the
schemes that result from implementing the random oracle by so called
"cryptographic hash functions". The main result of this paper is a negative
one: There exist signature and encryption schemes that are secure in the Random
Oracle Model, but for which any implementation of the random oracle results in
insecure schemes.
In the process of devising the above schemes, we consider possible
definitions for the notion of a "good implementation" of a random oracle,
pointing out limitations and challenges.Comment: 31 page
The Subgraph Testing Model
We initiate a study of testing properties of graphs that are presented as subgraphs of a fixed (or an explicitly given) graph. The tester is given free access to a base graph G=([n],E), and oracle access to a function f:E -> {0,1} that represents a subgraph of G. The tester is required to distinguish between subgraphs that posses a predetermined property and subgraphs that are far from possessing this property.
We focus on bounded-degree base graphs and on the relation between testing graph properties in the subgraph model and testing the same properties in the bounded-degree graph model. We identify cases in which testing is significantly easier in one model than in the other as well as cases in which testing has approximately the same complexity in both models. Our proofs are based on the design and analysis of efficient testers and on the establishment of query-complexity lower bounds
Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing
A graph is called self-ordered (a.k.a asymmetric) if the identity
permutation is its only automorphism. Equivalently, there is a unique
isomorphism from to any graph that is isomorphic to . We say that
is robustly self-ordered if the size of the symmetric difference
between and the edge-set of the graph obtained by permuting using any
permutation is proportional to the number of non-fixed-points of
. In this work, we initiate the study of the structure, construction and
utility of robustly self-ordered graphs.
We show that robustly self-ordered bounded-degree graphs exist (in
abundance), and that they can be constructed efficiently, in a strong sense.
Specifically, given the index of a vertex in such a graph, it is possible to
find all its neighbors in polynomial-time (i.e., in time that is
poly-logarithmic in the size of the graph).
We also consider graphs of unbounded degree, seeking correspondingly
unbounded robustness parameters. We again demonstrate that such graphs (of
linear degree) exist (in abundance), and that they can be constructed
efficiently, in a strong sense. This turns out to require very different tools.
Specifically, we show that the construction of such graphs reduces to the
construction of non-malleable two-source extractors (with very weak parameters
but with some additional natural features).
We demonstrate that robustly self-ordered bounded-degree graphs are useful
towards obtaining lower bounds on the query complexity of testing graph
properties both in the bounded-degree and the dense graph models. One of the
results that we obtain, via such a reduction, is a subexponential separation
between the query complexities of testing and tolerant testing of graph
properties in the bounded-degree graph model.Comment: Slightly modified and revised version of a CCC 2021 paper that also
appeared on ECCC 27: 149 (2020
Testing Distributions of Huge Objects
We initiate a study of a new model of property testing that is a hybrid of
testing properties of distributions and testing properties of strings.
Specifically, the new model refers to testing properties of distributions, but
these are distributions over huge objects (i.e., very long strings).
Accordingly, the model accounts for the total number of local probes into these
objects (resp., queries to the strings) as well as for the distance between
objects (resp., strings), and the distance between distributions is defined as
the earth mover's distance with respect to the relative Hamming distance
between strings.
We study the query complexity of testing in this new model, focusing on three
directions. First, we try to relate the query complexity of testing properties
in the new model to the sample complexity of testing these properties in the
standard distribution testing model. Second, we consider the complexity of
testing properties that arise naturally in the new model (e.g., distributions
that capture random variations of fixed strings). Third, we consider the
complexity of testing properties that were extensively studied in the standard
distribution testing model: Two such cases are uniform distributions and pairs
of identical distributions
Testing Distributions of Huge Objects
We initiate a study of a new model of property testing that is a hybrid of
testing properties of distributions and testing properties of strings.
Specifically, the new model refers to testing properties of distributions, but
these are distributions over huge objects (i.e., very long strings).
Accordingly, the model accounts for the total number of local probes into these
objects (resp., queries to the strings) as well as for the distance between
objects (resp., strings), and the distance between distributions is defined as
the earth mover's distance with respect to the relative Hamming distance
between strings.
We study the query complexity of testing in this new model, focusing on three
directions. First, we try to relate the query complexity of testing properties
in the new model to the sample complexity of testing these properties in the
standard distribution testing model. Second, we consider the complexity of
testing properties that arise naturally in the new model (e.g., distributions
that capture random variations of fixed strings). Third, we consider the
complexity of testing properties that were extensively studied in the standard
distribution testing model: Two such cases are uniform distributions and pairs
of identical distributions
Approximating Average Parameters of Graphs
Inspired by Feige (36th STOC, 2004), we initiate a study of sublinear randomized algorithms for approximating average parameters of a graph.
Specifically, we consider the average degree of a graph and the average distance between pairs of vertices in a graph.
Since our focus is on sublinear algorithms, these algorithms access the input graph via queries to an adequate oracle.
We consider two types of queries.
The first type is standard neighborhood queries (i.e., what is the i\u27th neighbor of vertex v?), whereas the second type are queries regarding the quantities that we need to find the average of (i.e., what is the degree of vertex v? and what is the distance between u and v, respectively).
Loosely speaking, our results indicate a difference between the two problems: For approximating the average degree, the standard neighbor queries suffice and in fact are preferable to degree queries. In contrast, for approximating average distances, the standard neighbor queries are of little help whereas distance queries are crucial
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Honest Verifier Statistical Zero-Knowledge Equals General Statistical Zero-Knowledge
We show how to transform any interactive proof system which is statistical zero-knowledge with respect to the honest-verifier, into a proof system which is statistical zero-knowledge with respect to any verifier. This is done by limiting the behavior of potentially cheating verifiers, without using computational assumptions or even referring to the complexity of such verifier strategies. (Previous transformations have either relied on computational assumptions or were applicable only to constant-round public-coin proof systems.)
Our transformation also applies to public-coin (aka Arthur-Merlin) computational zero-knowledge proofs: We transform any Arthur-Merlin proof system which is computational zero-knowledge with respect to the honest-verifier, into an Arthur-Merlin proof system which is computational zero-knowledge with respect to any probabilistic polynomial-time verifier.
A crucial ingredient in our analysis is a new lemma regarding 2-universal hashing functions.Engineering and Applied Science
On sample-based testers
The standard definition of property testing endows the tester with the ability to make arbitrary queries to “elements ” of the tested object. In contrast, sample-based testers only obtain independently distributed elements (a.k.a. labeled samples) of the tested object. While samplebased testers were defined by Goldreich, Goldwasser, and Ron (JACM 1998), most research in property testing is focused on query-based testers. In this work, we advance the study of sample-based property testers by providing several general positive results as well as by revealing relations between variants of this testing model. In particular: • We show that certain types of query-based testers yield sample-based testers of sublinear sample complexity. For example, this holds for a natural class of proximity oblivious testers. • We study the relation between distribution-free sample-based testers and one-sided error sample-based testers w.r.t the uniform distribution. While most of this work ignores the time complexity of testing, one part of it does focus on this aspect. The main result in this part is a sublinear-time sample-based tester for k-Colorability, for any k ≥ 2
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