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Optimal testing of ReedMuller codes
, 2009
"... We consider the problem of testing if a given function ..."
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Cited by 11 (7 self)
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We consider the problem of testing if a given function
A strong direct product theorem for disjointness
 In 42nd ACM Symposium on Theory of Computing (STOC
, 2010
"... A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication co ..."
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A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication complexity of the Disjointness problem, i.e., with communication const · kn the success probability of solving k instances can only be exponentially small in k. We show that this bound even holds in an AM communication protocol with limited ambiguity. The main result implies a new lower bound for Disjointness in a restricted 3player NOF protocol, and optimal communicationspace tradeoffs for Boolean matrix product. Our main result follows from a solution to the dual of a linear programming problem, whose feasibility comes from a socalled Intersection Sampling Lemma that generalizes a result by Razborov [Raz92]. We also discuss a new lower bound technique for randomized communication complexity called the generalized rectangle bound that we use in our proof. 1
STRONG DIRECT PRODUCT THEOREMS FOR QUANTUM COMMUNICATION AND QUERY COMPLEXITY
"... ABSTRACT. A strong direct product theorem (SDPT) states that solving n instances of a problem requires ˝.n / times the resources for a single instance, even to achieve success probability 2 ˝.n / : We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound fo ..."
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ABSTRACT. A strong direct product theorem (SDPT) states that solving n instances of a problem requires ˝.n / times the resources for a single instance, even to achieve success probability 2 ˝.n / : We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by the generalized discrepancy method, the strongest technique in that model. We prove that quantum query complexity obeys an SDPT whenever the query lower bound for a single instance is proved by the polynomial method, one of the two main techniques in that model. In both models, we prove the corresponding XOR lemmas and threshold direct product theorems. 1.
Shielding circuits with groups
, 2012
"... We show how to efficiently compile any given circuit C into a leakageresistant circuit Ĉ such that any function on the wires of Ĉ that leaks information during a computation Ĉ(x) yields advantage in computing the product of ĈΩ(1) elements of the alternating group Au. In combination with new compr ..."
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We show how to efficiently compile any given circuit C into a leakageresistant circuit Ĉ such that any function on the wires of Ĉ that leaks information during a computation Ĉ(x) yields advantage in computing the product of ĈΩ(1) elements of the alternating group Au. In combination with new compression bounds for Au products, also obtained here, Ĉ withstands leakage from virtually any class of functions against which averagecase lower bounds are known. This includes communication protocols, and AC 0 circuits augmented with few arbitrary symmetric gates. If NC 1 = TC 0 then the construction resists TC 0 leakage as well. We also conjecture that our construction resists NC 1 leakage. In addition, we extend the construction to the multiquery setting by relying on a simple secure hardware component. We build on Barrington’s theorem [JCSS ’89] and on the previous leakageresistant constructions by Ishai et al. [Crypto ’03] and Faust et al. [Eurocrypt ’10]. Our construction exploits properties of Au beyond what is sufficient for Barrington’s theorem.
The communication complexity of addition
, 2011
"... Suppose each of k ≤ no(1) players holds an nbit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a publiccoin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) bound by Nisan (Bolyai ..."
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Suppose each of k ≤ no(1) players holds an nbit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a publiccoin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) bound by Nisan (Bolyai Soc. Math. Studies; 1993). Our protocol also applies to addition modulo m. In this case we give a matching (publiccoin) Ω(k lg k) lower bound for various m. We also obtain some lower bounds over the integers, including Ω(k lg lg k) for protocols that are oneway, like ours. We give a protocol to determine if ∑ xi> s with error 1 % and communication O(k lg k) lg n. For k ≥ 3 this improves on Nisan’s O(k lg 2 n) bound. A similar improvement holds for computing degree(k − 1) polynomialthreshold functions in the numberonforehead model. We give a (publiccoin, 2player, tight) Ω(lg n) lower bound to determine if x1> x2. This improves on the Ω ( √ lg n) bound by Smirnov (1988).
Hard Functions for Lowdegree Polynomials over Prime Fields
"... In this paper, we present a new hardness amplification for lowdegree polynomials over prime fields, namely, we prove that if some function is mildly hard to approximate by any lowdegree polynomials then the sum of independent copies of the function is very hard to approximate by them. This result ..."
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In this paper, we present a new hardness amplification for lowdegree polynomials over prime fields, namely, we prove that if some function is mildly hard to approximate by any lowdegree polynomials then the sum of independent copies of the function is very hard to approximate by them. This result generalizes the XOR lemma for lowdegree polynomials over the binary field given by Viola and Wigderson [22]. The main technical contribution is the analysis of the Gowers norm over prime fields. For the analysis, we discuss a generalized lowdegree test, which we call the Gowers test, for polynomials over prime fields, which is a natural generalization of that over the binary field given by Alon, Kaufman, Krivelevich, Litsyn and Ron [2]. This Gowers test provides a new technique to analyze the Gowers norm over prime fields. Actually, the rejection probability of the Gowers test can be analyzed in the framework of Kaufman and Sudan [17]. However, our analysis is selfcontained and quantitatively better. By using our argument, we also prove the hardness of modulo functions for lowdegree polynomials over prime fields.
SPECIAL ISSUE: APPROXRANDOM 2012 A New Upper Bound on the Query Complexity of Testing Generalized ReedMuller Codes ∗
, 2012
"... Abstract: Over a finite field Fq, the (n,d,q)ReedMuller code is the code given by evaluations of nvariate polynomials of total degree at most d on all points (of Fn q). The task of testing if a function f: Fn q → Fq is close to a codeword of an (n,d,q)ReedMuller code has been of central interes ..."
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Abstract: Over a finite field Fq, the (n,d,q)ReedMuller code is the code given by evaluations of nvariate polynomials of total degree at most d on all points (of Fn q). The task of testing if a function f: Fn q → Fq is close to a codeword of an (n,d,q)ReedMuller code has been of central interest in complexity theory and property testing. The query complexity of this task is the minimal number of queries that a tester can make (minimum over all testers of the maximum number of queries over all random choices) while accepting all ReedMuller codewords and rejecting words that are δfar from the code with probability Ω(δ). (In this work we allow the constant in the Ω to depend on d.) For codes over a prime field Fq the optimal query complexity is wellknown and known to be Θ(q⌈(d+1)/(q−1) ⌉), and the test consists of testing if f is a degreed polynomial on a randomly chosen (⌈(d + 1)/(q − 1)⌉)dimensional affine subspace of Fn q. If q is not a prime,
Linear Systems Over Finite Abelian Groups
"... We consider a system of linear constraints over any finite Abelian group G of the following form: ℓi(x1,...,xn) ≡ ℓi,1x1 + ·· · + ℓi,nxn ∈ Ai for i = 1,...,t and each Ai ⊂ G, ℓi,j is an element of G and xi’s are Boolean variables. Our main result shows that the subset of the Boolean cube that satis ..."
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We consider a system of linear constraints over any finite Abelian group G of the following form: ℓi(x1,...,xn) ≡ ℓi,1x1 + ·· · + ℓi,nxn ∈ Ai for i = 1,...,t and each Ai ⊂ G, ℓi,j is an element of G and xi’s are Boolean variables. Our main result shows that the subset of the Boolean cube that satisfies these constraints has exponentially small correlation with the MODq boolean function, when the order of G and q are coprime numbers. Our work extends the recent result of Chattopadhyay and Wigderson (FOCS’09) who obtain such a correlation bound for linear systems over cyclic groups whose order is a product of two distinct primes or has at most one prime factor. Our result also immediately yields the first exponential boundsonthesizeofbooleandepthfour circuitsoftheformMAJ◦AND◦ANY O(1)◦ MODm for computing the MODq function, when m,q are coprime. No superpolynomial lower bounds were known for such circuits for computing any explicit function. This completely solves an open problem posed by Beigel and Maciel (Complexity’97). 1
www.theoryofcomputing.org Inverse Conjecture for the Gowers Norm is False
"... Abstract: Let p be a fixed prime number and N be a large integer. The “Inverse Conjecture for the Gowers norm ” states that if the “dth Gowers norm ” of a function f: F N p → Fp is nonnegligible, that is, larger than a constant independent of N, then f is nontrivially correlated to a degree(d − ..."
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Abstract: Let p be a fixed prime number and N be a large integer. The “Inverse Conjecture for the Gowers norm ” states that if the “dth Gowers norm ” of a function f: F N p → Fp is nonnegligible, that is, larger than a constant independent of N, then f is nontrivially correlated to a degree(d − 1) polynomial. The conjecture is known to hold for d = 2,3 and for any prime p. In this paper we show the conjecture to be false for p = 2 and d = 4, by presenting an explicit function whose 4th Gowers norm is nonnegligible, but whose correlation to any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao (2009). ACM Classification: F.2.2 AMS Classification: 05E99 Key words and phrases: Inverse Gowers conjecture, additive combinatorics, Gowers norm