Results 1  10
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19
On Yao’s XOR lemma
 Electronic Colloquium on Computational Complexity
, 1995
"... Abstract. A fundamental lemma of Yao states that computational weakunpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao’s Lemma and present a third alternative proof. The third proof proceeds by firs ..."
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Cited by 59 (7 self)
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Abstract. A fundamental lemma of Yao states that computational weakunpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao’s Lemma and present a third alternative proof. The third proof proceeds by first proving that a function constructed by concatenating the values of the original function on several independent instances is much more unpredictable, with respect to specified complexity bounds, than the original function. This statement turns out to be easier to prove than the XORLemma. Using a result of Goldreich and Levin (1989) and some elementary observation, we derive the XORLemma.
Quantum and Classical Strong Direct Product Theorems and Optimal TimeSpace Tradeoffs
 SIAM Journal on Computing
, 2004
"... 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 our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum ..."
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Cited by 44 (7 self)
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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 our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems...
P=BPP unless E has subexponential circuits: Derandomizing the XOR Lemma (Preliminary Version)
 In Proceedings of the 29th STOC
, 1996
"... Yao showed that the XOR of independent random instances of a somewhat hard Boolean function becomes almost completely unpredictable. In this paper we show that, in nonuniform settings, total independence is not necessary for this result to hold. We give a pseudorandom generator which produces n ..."
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Cited by 39 (6 self)
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Yao showed that the XOR of independent random instances of a somewhat hard Boolean function becomes almost completely unpredictable. In this paper we show that, in nonuniform settings, total independence is not necessary for this result to hold. We give a pseudorandom generator which produces n instances of the function for which the analog of the XOR lemma holds. This is the first derandomization of a "direct product" result. Our generator is a combination of two known ones  the random walks on expander graphs of [1, 9, 19] and the nearly disjoint subsets generator of [23]. The quality of the generator is proved via a new proof of the XOR lemma, which might also be useful for other direct product results. Combining our generator with the approach of [25, 6] and the generator of [16] gives substantially improved results for hardness vs. randomness tradeoffs. In particular, we show that if any problem in E = DT IME(2 O(n) ) has circuit complexity 2\Omega\Gamma n) , the...
The Cell Probe Complexity of Succinct Data Structures
 In Automata, Languages and Programming, 30th International Colloquium (ICALP 2003
, 2003
"... We show lower bounds in the cell probe model for the redundancy/query time tradeoff of solutions to static data structure problems. ..."
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Cited by 30 (0 self)
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We show lower bounds in the cell probe model for the redundancy/query time tradeoff of solutions to static data structure problems.
Towards Proving Strong Direct Product Theorems
 Computational Complexity
, 2001
"... A fundamental question of complexity theory is the direct product question. Namely weather the assumption that a function f is hard on average for some computational class (meaning that every algorithm from the class has small advantage over random guessing when computing f) entails that computin ..."
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Cited by 29 (1 self)
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A fundamental question of complexity theory is the direct product question. Namely weather the assumption that a function f is hard on average for some computational class (meaning that every algorithm from the class has small advantage over random guessing when computing f) entails that computing f on k independently chosen inputs is exponentially harder on average. A famous example is Yao's XORlemma, [Yao82] which gives such a result for boolean circuits. This question has also been studied in other computational models, such as decision trees [NRS94], and communication complexity [PRW97]. In Yao's XORlemma one assumes f is hard on average for circuits of size s and concludes that f #k (x 1 , , x k ) = f(x 1 ) # # f(x k ) is essentially exponentially harder on average for circuits of size s # . All known proofs of this lemma, [Lev85, Imp95, IW97, GNW95] have the feature that s # < s. In words, the circuit which attempts to compute f #k is smaller than the circuit whic...
Direct product results and the GCD problem, in old and new communication models
 In Proceedings of the 29th Annual ACM Symposium on Theory of Computing
, 1997
"... This paper contains several results regarding the communication complexity model and the 2prover games model, which are based on interaction between the two models: 1. We show how to improve the rate of exponential decrease in the parallel repetition theorem of [Ra] in terms of the communication co ..."
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Cited by 15 (1 self)
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This paper contains several results regarding the communication complexity model and the 2prover games model, which are based on interaction between the two models: 1. We show how to improve the rate of exponential decrease in the parallel repetition theorem of [Ra] in terms of the communication complexity of the verifier’s predicate. 2. We apply the improved parallel repetition theorem of 2prover games to derive, for the first time, a direct product theorem for communication complexity. The second derivation uses a common generalization of the two models, which is independently interesting. We initiate a study of its power by considering the GCD problem, and some variations of it, which exhibit a power gap between the new model and the classical communication complexity model. This gap is partly based on the following upper bounds: Given nbit inputs x and y to Alice and Bob respectively, they can achieve the tasks below with very high probability using only O(n / log n) communication bits:
Chernofftype Direct Product Theorems
 In Proceeding of the TwentySeventh Annual International Cryptology Conference (CRYPTO’07
, 2007
"... Abstract. Consider a challengeresponse protocol where the probability of a correct response is at least α for a legitimate user, and at most β < α for an attacker. One example is a CAPTCHA challenge, where a human should have a significantly higher chance of answering a single challenge (e.g., unco ..."
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Cited by 12 (5 self)
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Abstract. Consider a challengeresponse protocol where the probability of a correct response is at least α for a legitimate user, and at most β < α for an attacker. One example is a CAPTCHA challenge, where a human should have a significantly higher chance of answering a single challenge (e.g., uncovering a distorted letter) than an attacker; another example is an argument system without perfect completeness. A natural approach to boost the gap between legitimate users and attackers is to issue many challenges, and accept if the response is correct for more than a threshold fraction, for the threshold chosen between α and β. We give the first proof that parallel repetition with thresholds improves the security of such protocols. We do this with a very general result about an attacker’s ability to solve a large fraction of many independent instances of a hard problem, showing a Chernofflike convergence of the fraction solved incorrectly to the probability of failure for a single instance.
A Strong Direct Product Theorem for Corruption and theMultiparty NOF Communication Complexity of Disjointness
, 2005
"... We prove that twoparty randomized communication complexity satisfies a strong direct productproperty, so long as the communication lower bound is proved by a "corruption" or "onesided discrepancy" method over a rectangular distribution. We use this to prove new n\Omega (1) lower bounds for numbe ..."
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Cited by 8 (4 self)
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We prove that twoparty randomized communication complexity satisfies a strong direct productproperty, so long as the communication lower bound is proved by a "corruption" or "onesided discrepancy" method over a rectangular distribution. We use this to prove new n\Omega (1) lower bounds for numberontheforehead protocols in which the first player speaks once and then the other two players proceed arbitrarily. Using other techniques, we also establish an \Omega (n1/(k1)/(k 1)) lower bound for kplayerrandomized numberontheforehead protocols for the disjointness function in which all messages are broadcast simultaneously. A simple corollary of this is that general randomized numberontheforeheadprotocols require \Omega (log n/(k 1)) bits of communication to compute the disjointness 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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Cited by 7 (0 self)
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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
Quantum timespace tradeoffs for sorting
 Proceedings of 35th ACM STOC
, 2003
"... We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We o ..."
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Cited by 7 (1 self)
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We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We observe that for all storage bounds S, one can devise a quantum algorithm that sorts n numbers (using comparisons only) in time T = O(n