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39
Calibrating noise to sensitivity in private data analysis
 In Proceedings of the 3rd Theory of Cryptography Conference
, 2006
"... Abstract. We continue a line of research initiated in [10, 11] on privacypreserving statistical databases. Consider a trusted server that holds a database of sensitive information. Given a query function f mapping databases to reals, the socalled true answer is the result of applying f to the datab ..."
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Cited by 309 (47 self)
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Abstract. We continue a line of research initiated in [10, 11] on privacypreserving statistical databases. Consider a trusted server that holds a database of sensitive information. Given a query function f mapping databases to reals, the socalled true answer is the result of applying f to the database. To protect privacy, the true answer is perturbed by the addition of random noise generated according to a carefully chosen distribution, and this response, the true answer plus noise, is returned to the user. Previous work focused on the case of noisy sums, in which f =P i g(xi), where xi denotes the ith row of the database and g maps database rows to [0, 1]. We extend the study to general functions f, proving that privacy can be preserved by calibrating the standard deviation of the noise according to the sensitivity of the function f. Roughly speaking, this is the amount that any single argument to f can change its output. The new analysis shows that for several particular applications substantially less noise is needed than was previously understood to be the case. The first step is a very clean characterization of privacy in terms of indistinguishability of transcripts. Additionally, we obtain separation results showing the increased value of interactive sanitization mechanisms over noninteractive. 1 Introduction We continue a line of research initiated in [10, 11] on privacy in statistical databases. A statistic is a quantity computed from a sample. Intuitively, if the database is a representative sample of an underlying population, the goal ofa privacypreserving statistical database is to enable the user to learn properties of the population as a whole while protecting the privacy of the individualcontributors.
Robust PCPs of Proximity, Shorter PCPs and Applications to Coding
 in Proc. 36th ACM Symp. on Theory of Computing
, 2004
"... We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) ..."
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Cited by 80 (25 self)
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We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) Boolean queries.
A combinatorial characterization of the testable graph properties: it’s all about regularity
 Proc. of STOC 2006
, 2006
"... A common thread in all the recent results concerning testing dense graphs is the use of Szemerédi’s regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédipartition is testable with a constant ..."
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Cited by 69 (14 self)
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A common thread in all the recent results concerning testing dense graphs is the use of Szemerédi’s regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédipartition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property P can be tested with a constant number of queries if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemerédipartitions. This means that in some sense, testing for Szemerédipartitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of propertytesting, which was first raised in the 1996 paper of Goldreich, Goldwasser and Ron [24] that initiated the study of graph propertytesting. This characterization also gives an intuitive explanation as to what makes a graph property testable.
Algebraic Property Testing: The Role of Invariance
, 2007
"... We argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space K n over a field K to a subfield F. We consider Flinear properties that are i ..."
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Cited by 35 (16 self)
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We argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space K n over a field K to a subfield F. We consider Flinear properties that are invariant under linear transformations of the domain and prove that an O(1)local “characterization ” is a necessary and sufficient condition for O(1)local testability when K  = O(1). (A local characterization of a property is a definition of a property in terms of local constraints satisfied by functions exhibiting a property.) For the subclass of properties that are invariant under affine transformations of the domain, we prove that the existence of a single O(1)local constraint implies O(1)local testability. These results generalize and extend the class of algebraic properties, most notably linearity and lowdegreeness, that were previously known to be testable. In particular, the extensions include properties satisfied by functions of degree linear in n that turn out to be O(1)locally testable. Our results are proved by introducing a new notion that we term “formal characterizations”. Roughly this corresponds to characterizations that are given by a single local constraint and its permutations under linear transformations of the domain. Our main testing result shows that local formal characterizations
Functions That Have ReadTwice Constant Width Branching Programs Are Not Necessarily Testable
, 2003
"... ..."
Robust Locally Testable Codes and Products of Codes
 In Proc. RANDOM: International Workshop on Randomization and Approximation Techniques in Computer Science
, 2004
"... We continue the investigation of locally testable codes, i.e., errorcorrecting codes for whom membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed ..."
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Cited by 18 (6 self)
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We continue the investigation of locally testable codes, i.e., errorcorrecting codes for whom membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed notion of robust probabilistically checkable proofs, we introduce the notion of robust local testability of codes. We relate this notion to a product of codes introduced by Tanner, and show a very simple composition lemma for this notion. Next, we show that codes built by tensor products can be tested robustly and somewhat locally, by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing lowdegree multivariate polynomials (which are a special case of tensor codes).
2transitivity is insufficient for local testability
 In CCC 2008: Proceedings of the 23rd IEEE Conference on Computational Complexity
, 2008
"... A basic goal in Property Testing is to identify a minimal set of features that make a property testable. For the case when the property to be tested is membership in a binary linear errorcorrecting code, Alon et al. [2] had conjectured that the presence of a single low weight code in the dual, and ..."
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Cited by 16 (12 self)
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A basic goal in Property Testing is to identify a minimal set of features that make a property testable. For the case when the property to be tested is membership in a binary linear errorcorrecting code, Alon et al. [2] had conjectured that the presence of a single low weight code in the dual, and “2transitivity ” of the code (i.e., the code is invariant under a 2transitive group of permutations on the coordinates of the code) suffice to get local testability. We refute this conjecture by giving a family of error correcting codes where the coordinates of the codewords form a large field of characteristic two, and the code is invariant under affine transformations of the domain. This class of properties was introduced by Kaufman and Sudan [13] as a setting where many results in algebraic property testing generalize. Our result shows a complementary virtue: this family also can be useful in producing counterexamples to natural conjectures. 1
Bounds on 2Query Codeword Testing
 IN THE PROCEEDINGS OF RANDOM'03, SPRINGER LNCS
, 2003
"... We present upper bounds on the size of codes that are locally testable by querying only two input symbols. For linear codes, we show that any 2locally testable code with minimal distance ffin over a finite field F cannot have more than jFj codewords. This result holds even for testers with tw ..."
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Cited by 13 (4 self)
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We present upper bounds on the size of codes that are locally testable by querying only two input symbols. For linear codes, we show that any 2locally testable code with minimal distance ffin over a finite field F cannot have more than jFj codewords. This result holds even for testers with twosided error. For general (nonlinear) codes we obtain the exact same bounds on the code size as a function of the minimal distance, but our bounds apply only for binary alphabets and onesided error testers (i.e. with perfect completeness). Our bounds are obtained by examining the graph induced by the set of possible pairs of queries made by a codeword tester on a given code. We also demonstrate the tightness of our upper bounds and the essential role of certain parameters.
Tolerant versus intolerant testing for boolean properties
 In Proceedings of the 20th IEEE Conference on Computational Complexity
, 2005
"... Abstract: A property tester with high probability accepts inputs satisfying a given property and rejects inputs that are far from satisfying it. A tolerant property tester, as defined by Parnas, Ron and Rubinfeld, must also accept inputs that are close enough to satisfying the property. We construct ..."
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Cited by 13 (2 self)
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Abstract: A property tester with high probability accepts inputs satisfying a given property and rejects inputs that are far from satisfying it. A tolerant property tester, as defined by Parnas, Ron and Rubinfeld, must also accept inputs that are close enough to satisfying the property. We construct two properties of binary functions for which there exists a test making a constant number of queries, but yet there exists no such tolerant test. The first construction uses Hadamard codes and long codes. Then, using Probabilistically Checkable Proofs of Proximity as constructed by BenSasson et al., we exhibit a property which has constant query intolerant testers but for which any tolerant tester requires n Ω(1) queries. ACM Classification: G.3, F.2.2 AMS Classification: 68Q99, 68W20
Limits on the rate of locally testable affineinvariant codes
, 2009
"... Despite its many applications, to program checking, probabilistically checkable proofs, locally testable and locally decodable codes, and cryptography, “algebraic property testing ” is not wellunderstood. A significant obstacle to a better understanding, was a lack of a concrete definition that abst ..."
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Cited by 12 (8 self)
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Despite its many applications, to program checking, probabilistically checkable proofs, locally testable and locally decodable codes, and cryptography, “algebraic property testing ” is not wellunderstood. A significant obstacle to a better understanding, was a lack of a concrete definition that abstracted known testable algebraic properties and reflected their testability. This obstacle was removed by [Kaufman and Sudan, STOC 2008] who considered (linear) “affineinvariant properties”, i.e., properties that are closed under summation, and under affine transformations of the domain. Kaufman and Sudan showed that these two features (linearity of the property and its affineinvariance) play a central role in the testability of many known algebraic properties. However their work does not give a complete characterization of the testability of affineinvariant properties, and several technical obstacles need to be overcome to obtain such a characterization. Indeed, their work left open the tantalizing possibility that locally testable codes of rate dramatically better than that of the family of ReedMuller codes (the most popular form of locally testable codes, which also happen to be affineinvariant) could be found by systematically exploring the space of affineinvariant properties.