Abstract. Informally, an obfuscator O is an (ecient, probabilistic) \\compiler " that takes as input a program (or circuit) P and produces a new program O(P) that has the same functionality as P yet is \\unintel-ligible " in some sense. Obfuscators, if they exist, would have a wide vari-ety of cryptographic and complexity-theoretic applications, ranging from software protection to homomorphic encryption to complexity-theoretic analogues of Rice’s theorem. Most of these applications are based on an interpretation of the \\unintelligibility " condition in obfuscation as mean-ing that O(P) is a \\virtual black box, " in the sense that anything one can eciently compute given O(P), one could also eciently compute given oracle access to P. In this work, we initiate a theoretical investigation of obfuscation. Our main result is that, even under very weak formalizations of the above in-tuition, obfuscation is impossible. We prove this by constructing a family of functions F that are inherently unobfuscatable in the following sense:

Entanglement is an essential resource for quantum communication and quantum computation, similar to shared random bits in the classical world. Entanglement distillation extracts nearly-perfect entanglement from imperfect entangled state. The classical communication complexity of these protocols is the minimal amount of classical information that needs to be exchanged for the conversion. In this paper, we focus on the communication complexity of protocols that operate with incomplete information, i.e., where the inputs are mixed states and/or prepared adversarially. We consider three models of imperfect entanglement, namely, the bounded measurement model, the depolarization model, and the fidelity model. We describe these models as well as the motivations for studying them. For the bounded measurement model and the depolarization model, we prove tight and almost-tight bounds on the output quality of non-interactive protocols. For the fidelity model, we prove a lower bound that matches the upper bound given by Ambainis et al., and thus completely characterizes communication complexity of entanglement distillation protocols for this model. Our result also suggests the optimality of the BB84 protocol in terms of communication complexity. We emphasize that although some of the results appear intuitively straightforward, their proofs are not. In fact, two novel techniques are developed for proving these results. We believe that these techniques are of independent interests, too.

This paper is about a variant of k nearest neighbor classification on large many-class high dimensional datasets.K nearest neighbor remains a popular classification technique, especially in areas such as computer vision, drug activity prediction and astrophysics. Furthermore, many more modern classifiers, such as kernel-based Bayes classifiers or the prediction phase of SVMs, require computational regimes similar to k-NN. We believe that tractable k-NN algorithms therefore continue to be important.This paper relies on the insight that even with many classes, the task of finding the majority class among the k nearest neighbors of a query need not require us to explicitly find those k nearest neighbors. This insight was previously used in (Liu et al., 2003) in two algorithms called KNS2 and KNS3 which dealt with fast classification in the case of two classes. In this paper we show how a different approach, IOC (standing for the International Olympic Committee) can apply to the case of n classes where n > 2.IOC assumes a slightly different processing of the datapoints in the neighborhood of the query. This allows it to search a set of metric trees, one for each class. During the searches it is possible to quickly prune away classes that cannot possibly be the majority.We give experimental results on datasets of up to 5.8 x 105 records and 1.5 x 103 attributes, frequently showing an order of magnitude acceleration compared with each of (i) conventional linear scan, (ii) a well-known independent SR-tree implementation of conventional k-NN and (iii) a highly optimized conventional k-NN metric tree search.

This paper concerns approximate nearest neighbor searching algorithms, which have become increasingly important, especially in high dimen- sional perception areas such as computer vision, with dozens of publica- tions in recent years. Much of this enthusiasm is due to a successful new approximate nearest neighbor approach called Locality Sensitive Hash- ing (LSH). In this paper we ask the question: can earlier spatial data structure approaches to exact nearest neighbor, such as metric trees, be altered to provide approximate answers to proximity queries and if so, how? We introduce a new kind of metric tree that allows overlap: certain datapoints may appear in both the children of a parent. We also intro- duce new approximate k-NN search algorithms on this structure. We show why these structures should be able to exploit the same random- projection-based approximations that LSH enjoys, but with a simpler al- gorithm and perhaps with greater efficiency. We then provide a detailed empirical evaluation on five large, high dimensional datasets which show up to 31-fold accelerations over LSH. This result holds true throughout the spectrum of approximation levels.

We study the problem of extracting Einstein-Podolsky-Rosen (EPR) pairs from a general source of entanglement. Suppose Alice and Bob share a bipartite state /spl rho/ which is "reasonably close" to perfect EPR pairs. The only information Alice and Bob possess is a lower bound on the fidelity of /spl rho/ and a maximally entangled state. They wish to "purify" /spl rho/ using local operations and classical communication, and output a state that is arbitrarily close to EPR pairs. We prove that, on average, Alice and Bob cannot increase the fidelity of the input state significantly. On the other hand, there exist protocols that may fail with a small probability, and otherwise will output states arbitrarily close to EPR pairs with very high probability. These protocols come from the "purity-testing protocols" of H. Barnum et al. (2001).