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Property Testing Lower Bounds Via Communication Complexity
, 2011
"... We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexit ..."
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We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is klinear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with twosided error, thus confirming a conjecture of Goldreich [25]. The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as kjuntas. For some classes, such as the class of monotone functions and the class of ssparse GF(2) polynomials, we significantly strengthen the best known bounds.
Untestable properties in the KahrMooreWang class
 in Proc. WoLLIC 2011, ser. Lecture Notes in Computer Science, L. Beklemishev and R. de Queiroz, Eds
, 2011
"... Abstract. Property testing is a kind of randomized approximation in which one takes a small, random sample of a structure and wishes to determine whether the structure satisfies some property or is far from satisfying the property. We focus on the testability of classes of firstorder expressible pr ..."
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Abstract. Property testing is a kind of randomized approximation in which one takes a small, random sample of a structure and wishes to determine whether the structure satisfies some property or is far from satisfying the property. We focus on the testability of classes of firstorder expressible properties, and in particular, on the classification of prefixvocabulary classes for testability. The main result is the untestability of [∀∃∀, (0, 1)]=. This is a wellknown class and minimal for untestability. We discuss what is currently known about the classification for testability and briefly compare it to other classifications. Key words: property testing, logic, randomized algorithms 1
Local correction of juntas
 Information Processing Letters, Volume 112, Issue 6
, 2012
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GraphTheoretic Algorithms for the “Isomorphism of Polynomials ” Problem
"... Abstract. We give three new algorithms to solve the “isomorphism of polynomial ” problem, which was underlying the hardness of recovering the secretkey in some multivariate trapdoor oneway functions. In this problem, the adversary is given two quadratic functions, with the promise that they are eq ..."
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Abstract. We give three new algorithms to solve the “isomorphism of polynomial ” problem, which was underlying the hardness of recovering the secretkey in some multivariate trapdoor oneway functions. In this problem, the adversary is given two quadratic functions, with the promise that they are equal up to linear changes of coordinates. Her objective is to compute these changes of coordinates, a task which is known to be harder than GraphIsomorphism. Our new algorithm build on previous work in a novel way. Exploiting the birthday paradox, we break instances of the problem in time q 2n/3 (rigorously) and q n/2 (heuristically), where q n is the time needed to invert the quadratic trapdoor function by exhaustive search. These results are obtained by turning the algebraic problem into a combinatorial one, namely that of recovering partial information on an isomorphism between two exponentially large graphs. These graphs, derived from the quadratic functions, are new tools in multivariate cryptanalysis. 1
Tight Lower Bounds for Testing Linear Isomorphism
"... We study lower bounds for testing membership in families of linear/affineinvariant Boolean functions over the hypercube. A family of functions P ⊆ {{0, 1} n → {0, 1}} is linear/affine invariant if for any f ∈ P, it is the case that f ◦ L ∈ P for any linear/affine transformation L of the domain. Mot ..."
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We study lower bounds for testing membership in families of linear/affineinvariant Boolean functions over the hypercube. A family of functions P ⊆ {{0, 1} n → {0, 1}} is linear/affine invariant if for any f ∈ P, it is the case that f ◦ L ∈ P for any linear/affine transformation L of the domain. Motivated by the recent resurgence of attention to the permutation isomorphism problem, we first focus on families that are linearly/affinely isomorphic to some fixed function. A function f: {0, 1} n → {0, 1} is called linear isomorphic to a fixed Boolean function g if f = g ◦ A for some nonsingular transformation A. Our main result is a tight adaptive, twosided Ω(n2) lower bound for testing linear isomorphism to the innerproduct function. This is the first lower bound for testing linear isomorphism to a specific function that matches the trivial upper bound. Our proof exploits the elegant connection between testing and communication complexity discovered by Blais et al. (Computational Complexity, 2012.) Our results are also the first instance of this connection that gives better than Ω(n) lower bound for any property of Boolean functions. These results extend to testing linear isomorphism to any fixed function in the larger class of socalled MaioranaMcFarland bent functions. Our second result shows an Ω(2n/4) query lower bound for any adaptive, twosided tester for membership in the MaioranaMcFarland class of bent functions. This class of Boolean functions is also affineinvariant and its rich structure and pseudorandom properties have been wellstudied in mathematics, coding theory and cryptography.
Local Correction with Constant Error Rate
, 2012
"... A Boolean function f of n variables is said to be qlocally correctable if, given a blackbox access to a function g which is ”close” to an isomorphism fσ(x) = fσ(x1,...,xn) = f(x σ(1),...,x σ(n)) of f, we can compute fσ(x) for any x ∈ Z n 2 with good probability using q queries to g. It is known th ..."
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A Boolean function f of n variables is said to be qlocally correctable if, given a blackbox access to a function g which is ”close” to an isomorphism fσ(x) = fσ(x1,...,xn) = f(x σ(1),...,x σ(n)) of f, we can compute fσ(x) for any x ∈ Z n 2 with good probability using q queries to g. It is known that degree d polynomials are O(2 d)locally correctable, and that most kjuntas are O(k log k)locally correctable, where the closeness parameter, or more precisely the distance between g and fσ, is required to be exponentially small (in d and k respectively). In this work we relax the requirement for the closeness parameter by allowing the distance between the functions to be a constant. We first investigate the family of juntas, and show that almost every kjunta is O(k log² k)locally correctable for any distance ε < 0.001. A similar result is shown for the family of partially symmetric functions, that is functions which are indifferent to any reordering of all but a constant number of their variables. For both families, the algorithms provided here use nonadaptive queries and are applicable to most but not all functions of each family (as it is shown to be impossible to locally correct all of them). Our approach utilizes the measure of symmetric influence introduced in the recent analysis of testing partial symmetry of functions.
“Isomorphism of Polynomials ” Problem
, 2013
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Recent Progress in the Classification for Testability
"... Abstract—In property testing we wish to distinguish between objects that have a given property and objects that are far from having the property, after examining only a small portion of the object. The classification problem for testability focuses on properties that are expressible in sentences of ..."
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Abstract—In property testing we wish to distinguish between objects that have a given property and objects that are far from having the property, after examining only a small portion of the object. The classification problem for testability focuses on properties that are expressible in sentences of firstorder logic, and asks us to determine which prefix vocabulary classes of firstorder logic are testable and which are not. Here we review recent results concerning this classification. I.
00002012 to 00002012 4. TITLE AND SUBTITLE
, 2012
"... and the Fonds de recherche du Québec – Nature et technologies. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any oth ..."
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and the Fonds de recherche du Québec – Nature et technologies. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. Report Documentation Page Form ApprovedOMB No. 07040188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 222024302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it