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 XOR-Lemma. Using a result of Goldreich and Levin (1989) and some elementary observation, we derive the XOR-Lemma.

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...

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 non-uniform settings, total independence is not necessary for this result to hold. We give a pseudo-random 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 trade-offs. In particular, we show that if any problem in E = DT IME(2 O(n) ) has circuit complexity 2\Omega\Gamma n) , the...

We show lower bounds in the cell probe model for the redundancy/query time tradeoff of solutions to static data structure problems.

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 XOR-lemma, [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 XOR-lemma 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...

This paper contains several results regarding the communication complexity model and the 2-prover 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 2-prover 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 n-bit 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) communica...

Abstract. Consider a challenge-response 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 Chernoff-like convergence of the fraction solved incorrectly to the probability of failure for a single instance.

We prove that two-party randomized communication complexity satisfies a strong direct productproperty, so long as the communication lower bound is proved by a "corruption" or "one-sided discrepancy" method over a rectangular distribution. We use this to prove new n\Omega (1) lower bounds for number-on-the-forehead 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/(k-1)/(k- 1)) lower bound for k-playerrandomized number-on-the-forehead protocols for the disjointness function in which all messages are broadcast simultaneously. A simple corollary of this is that general randomized number-on-the-foreheadprotocols require \Omega (log n/(k- 1)) bits of communication to compute the disjointness function.

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

We prove that corruption, one of the most powerful measures used to analyze 2-party randomized communication complexity, satisfies a strong direct sum property under rectangular distributions. This direct sum bound holds even when the error is allowed to be exponentially close to 1. We use this to analyze the complexity of the widelystudied set disjointness problem in the usual “number-onthe-forehead” (NOF) model of multiparty communication complexity. 1