We show how to build indistinguishability obfuscation (iO) for Turing Machines where the overhead is polynomial in the security parameter λ, machine description |M| and input size |x| (with only a negligible correctness error). In particular, we avoid growing polynomially with the maximum space of a computation. Our construction is based on iO for circuits, one way functions and injective pseudo random generators. Our results are based on new "selective enforcement" techniques. Here we first create a primitive called positional accumulators that allows for a small commitment to a much larger storage. The commitment is unconditionally sound for a select piece of the storage. This primitive serves as an "iO-friendly" tool that allows us to make two different programs equivalent at different stages of a proof. The pieces of storage that are selected depend on what hybrid stage we are at in a proof. We first build up our enforcement ideas in a simpler context of "message hiding encodings" and work our way up to indistinguishability obfuscation.
{"title":"Indistinguishability Obfuscation for Turing Machines with Unbounded Memory","authors":"Venkata Koppula, Allison Bishop, Brent Waters","doi":"10.1145/2746539.2746614","DOIUrl":"https://doi.org/10.1145/2746539.2746614","url":null,"abstract":"We show how to build indistinguishability obfuscation (iO) for Turing Machines where the overhead is polynomial in the security parameter λ, machine description |M| and input size |x| (with only a negligible correctness error). In particular, we avoid growing polynomially with the maximum space of a computation. Our construction is based on iO for circuits, one way functions and injective pseudo random generators. Our results are based on new \"selective enforcement\" techniques. Here we first create a primitive called positional accumulators that allows for a small commitment to a much larger storage. The commitment is unconditionally sound for a select piece of the storage. This primitive serves as an \"iO-friendly\" tool that allows us to make two different programs equivalent at different stages of a proof. The pieces of storage that are selected depend on what hybrid stage we are at in a proof. We first build up our enforcement ideas in a simpler context of \"message hiding encodings\" and work our way up to indistinguishability obfuscation.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76480959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a novel technique which enables two players to maintain an estimate of the internal information cost of their conversation in an online fashion without revealing much extra information. We use this construction to obtain new results about communication complexity and information-theoretic privacy. As a first corollary, we prove a strong direct product theorem for communication complexity in terms of information complexity: If I bits of information are required for solving a single copy of f under μ with probability 2/3, then any protocol attempting to solve n independent copies of f under μn using o(n • I) communication, will succeed with probability 2-Ω(n). This is tight, as Braverman and Rao [BR11] previously showed that O(n • I) communication suffice to succeed with probability ~(2/3)n. We then show how the information odometer can be used to achieve the best possible information-theoretic privacy between two untrusted parties: If the players' goal is to compute a function f(x,y), and f admits a protocol with information cost is I and communication cost C, then our odometer can be used to produce a "robust" protocol which: (i) Assuming both players are honest, computes f with high probability, and (ii) Even if one party is malicious, then for any k∈N, the probability that the honest player reveals more than O(k • (I+log C)) bits of information to the other player is at most 2-Ω(k). Finally, we outline an approach which uses the odometer as a proxy for breaking state of the art interactive compression results: We show that our odometer allows to reduce interactive compression to the regime where I=O(log C), thereby opening a potential avenue for improving the compression result of [BBCR10] and to new direct sum and product theorems in communication complexity.
{"title":"An Interactive Information Odometer and Applications","authors":"M. Braverman, Omri Weinstein","doi":"10.1145/2746539.2746548","DOIUrl":"https://doi.org/10.1145/2746539.2746548","url":null,"abstract":"We introduce a novel technique which enables two players to maintain an estimate of the internal information cost of their conversation in an online fashion without revealing much extra information. We use this construction to obtain new results about communication complexity and information-theoretic privacy. As a first corollary, we prove a strong direct product theorem for communication complexity in terms of information complexity: If I bits of information are required for solving a single copy of f under μ with probability 2/3, then any protocol attempting to solve n independent copies of f under μn using o(n • I) communication, will succeed with probability 2-Ω(n). This is tight, as Braverman and Rao [BR11] previously showed that O(n • I) communication suffice to succeed with probability ~(2/3)n. We then show how the information odometer can be used to achieve the best possible information-theoretic privacy between two untrusted parties: If the players' goal is to compute a function f(x,y), and f admits a protocol with information cost is I and communication cost C, then our odometer can be used to produce a \"robust\" protocol which: (i) Assuming both players are honest, computes f with high probability, and (ii) Even if one party is malicious, then for any k∈N, the probability that the honest player reveals more than O(k • (I+log C)) bits of information to the other player is at most 2-Ω(k). Finally, we outline an approach which uses the odometer as a proxy for breaking state of the art interactive compression results: We show that our odometer allows to reduce interactive compression to the regime where I=O(log C), thereby opening a potential avenue for improving the compression result of [BBCR10] and to new direct sum and product theorems in communication complexity.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88341544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Divesh Aggarwal, Y. Dodis, Tomasz Kazana, Maciej Obremski
Non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs [DPW10], provide a useful message integrity guarantee in situations where traditional error-correction (and even error-detection) is impossible; for example, when the attacker can completely overwrite the encoded message. Informally, a code is non-malleable if the message contained in a modified codeword is either the original message, or a completely "unrelated value". Although such codes do not exist if the family of "tampering functions" cF allowed to modify the original codeword is completely unrestricted, they are known to exist for many broad tampering families cF. The family which received the most attention [DPW10,LL12,DKO13,ADL14,CG14a,CG14b] is the family of tampering functions in the so called (2-part) split-state model: here the message x is encoded into two shares L and R, and the attacker is allowed to arbitrarily tamper with each L and R individually. Despite this attention, the following problem remained open: Build efficient, information-theoretically secure non-malleable codes in the split-state model with constant encoding rate: |L|=|R|=O(|x|). In this work, we resolve this open problem. Our technique for getting our main result is of independent interest. We develop a generalization of non-malleable codes, called non-malleable reductions; show simple composition theorem for non-malleable reductions; build a variety of such reductions connecting various (independently interesting) tampering families cF to each other; construct several new non-malleable codes in the split-state model by applying the composition theorem to a series of easy to understand reductions. Most importantly, we show several "independence amplification" reductions, showing how to reduce split-state tampering of very few parts to an easier question of split-state tampering with a much larger number of parts. In particular, our final, constant-rate, non-malleable code composes one of these reductions with the very recent, "9-split-state" code of Chattopadhyay and Zuckerman [CZ14].
{"title":"Non-malleable Reductions and Applications","authors":"Divesh Aggarwal, Y. Dodis, Tomasz Kazana, Maciej Obremski","doi":"10.1145/2746539.2746544","DOIUrl":"https://doi.org/10.1145/2746539.2746544","url":null,"abstract":"Non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs [DPW10], provide a useful message integrity guarantee in situations where traditional error-correction (and even error-detection) is impossible; for example, when the attacker can completely overwrite the encoded message. Informally, a code is non-malleable if the message contained in a modified codeword is either the original message, or a completely \"unrelated value\". Although such codes do not exist if the family of \"tampering functions\" cF allowed to modify the original codeword is completely unrestricted, they are known to exist for many broad tampering families cF. The family which received the most attention [DPW10,LL12,DKO13,ADL14,CG14a,CG14b] is the family of tampering functions in the so called (2-part) split-state model: here the message x is encoded into two shares L and R, and the attacker is allowed to arbitrarily tamper with each L and R individually. Despite this attention, the following problem remained open: Build efficient, information-theoretically secure non-malleable codes in the split-state model with constant encoding rate: |L|=|R|=O(|x|). In this work, we resolve this open problem. Our technique for getting our main result is of independent interest. We develop a generalization of non-malleable codes, called non-malleable reductions; show simple composition theorem for non-malleable reductions; build a variety of such reductions connecting various (independently interesting) tampering families cF to each other; construct several new non-malleable codes in the split-state model by applying the composition theorem to a series of easy to understand reductions. Most importantly, we show several \"independence amplification\" reductions, showing how to reduce split-state tampering of very few parts to an easier question of split-state tampering with a much larger number of parts. In particular, our final, constant-rate, non-malleable code composes one of these reductions with the very recent, \"9-split-state\" code of Chattopadhyay and Zuckerman [CZ14].","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82653653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function f:Z+→ Z+, such that for any integer g> 0, any graph of treewidth at least f(g), contains the (g x g)-grid as a minor. Until recently, the best known upper bounds on f were super-exponential in g. A recent work of Chekuri and Chuzhoy provided the first polynomial bound, by showing that treewidth f(g)=O(g98 poly log g) is sufficient to ensure the existence of the (g x g)-grid minor in any graph. In this paper we provide a much simpler proof of the Excluded Grid Theorem, achieving a bound of $f(g)=O(g^{36} poly log g)$. Our proof is self-contained, except for using prior work to reduce the maximum vertex degree of the input graph to a constant.
研究了Robertson和Seymour的排除网格定理。这是图论中的一个基本结果，它表明存在某个函数f:Z+→Z+，使得对于任何整数g> 0，任何树宽至少为f(g)的图，都包含(g x g)-网格作为次要项。直到最近，最著名的f的上界是g的超指数上界。Chekuri和Chuzhoy最近的一项工作提供了第一个多项式边界，通过证明树宽度f(g)=O(g98 poly log g)足以确保在任何图中存在(g x g)-网格。在本文中，我们提供了一个更简单的排除网格定理的证明，得到$f(g)=O(g^{36} poly log g)$的界。我们的证明是自包含的，除了使用先前的工作将输入图的最大顶点度降低到一个常数。
{"title":"Excluded Grid Theorem: Improved and Simplified","authors":"Julia Chuzhoy","doi":"10.1145/2746539.2746551","DOIUrl":"https://doi.org/10.1145/2746539.2746551","url":null,"abstract":"We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function f:Z+→ Z+, such that for any integer g> 0, any graph of treewidth at least f(g), contains the (g x g)-grid as a minor. Until recently, the best known upper bounds on f were super-exponential in g. A recent work of Chekuri and Chuzhoy provided the first polynomial bound, by showing that treewidth f(g)=O(g98 poly log g) is sufficient to ensure the existence of the (g x g)-grid minor in any graph. In this paper we provide a much simpler proof of the Excluded Grid Theorem, achieving a bound of $f(g)=O(g^{36} poly log g)$. Our proof is self-contained, except for using prior work to reduce the maximum vertex degree of the input graph to a constant.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84159276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An effective technique for solving optimization problems over massive data sets is to partition the data into smaller pieces, solve the problem on each piece and compute a representative solution from it, and finally obtain a solution inside the union of the representative solutions for all pieces. This technique can be captured via the concept of composable core-sets, and has been recently applied to solve diversity maximization problems as well as several clustering problems [7,15,8]. However, for coverage and submodular maximization problems, impossibility bounds are known for this technique [15]. In this paper, we focus on efficient construction of a randomized variant of composable core-sets where the above idea is applied on a random clustering of the data. We employ this technique for the coverage, monotone and non-monotone submodular maximization problems. Our results significantly improve upon the hardness results for non-randomized core-sets, and imply improved results for submodular maximization in a distributed and streaming settings. The effectiveness of this technique has been confirmed empirically for several machine learning applications [22], and our proof provides a theoretical foundation to this idea. In summary, we show that a simple greedy algorithm results in a 1/3-approximate randomized composable core-set for submodular maximization under a cardinality constraint. Our result also extends to non-monotone submodular functions, and leads to the first 2-round MapReduce-based constant-factor approximation algorithm with O(n) total communication complexity for either monotone or non-monotone functions. Finally, using an improved analysis technique and a new algorithm PseudoGreedy, we present an improved 0.545-approximation algorithm for monotone submodular maximization, which is in turn the first MapReduce-based algorithm beating factor 1/2 in a constant number of rounds.
{"title":"Randomized Composable Core-sets for Distributed Submodular Maximization","authors":"V. Mirrokni, Morteza Zadimoghaddam","doi":"10.1145/2746539.2746624","DOIUrl":"https://doi.org/10.1145/2746539.2746624","url":null,"abstract":"An effective technique for solving optimization problems over massive data sets is to partition the data into smaller pieces, solve the problem on each piece and compute a representative solution from it, and finally obtain a solution inside the union of the representative solutions for all pieces. This technique can be captured via the concept of composable core-sets, and has been recently applied to solve diversity maximization problems as well as several clustering problems [7,15,8]. However, for coverage and submodular maximization problems, impossibility bounds are known for this technique [15]. In this paper, we focus on efficient construction of a randomized variant of composable core-sets where the above idea is applied on a random clustering of the data. We employ this technique for the coverage, monotone and non-monotone submodular maximization problems. Our results significantly improve upon the hardness results for non-randomized core-sets, and imply improved results for submodular maximization in a distributed and streaming settings. The effectiveness of this technique has been confirmed empirically for several machine learning applications [22], and our proof provides a theoretical foundation to this idea. In summary, we show that a simple greedy algorithm results in a 1/3-approximate randomized composable core-set for submodular maximization under a cardinality constraint. Our result also extends to non-monotone submodular functions, and leads to the first 2-round MapReduce-based constant-factor approximation algorithm with O(n) total communication complexity for either monotone or non-monotone functions. Finally, using an improved analysis technique and a new algorithm PseudoGreedy, we present an improved 0.545-approximation algorithm for monotone submodular maximization, which is in turn the first MapReduce-based algorithm beating factor 1/2 in a constant number of rounds.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87298647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present algorithmic applications of an approximate version of Caratheodory's theorem. The theorem states that given a set of vectors X in Rd, for every vector in the convex hull of X there exists an ε-close (under the p-norm distance, for 2 ≤ p < ∞) vector that can be expressed as a convex combination of at most b vectors of X, where the bound b depends on ε and the norm p and is independent of the dimension d. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pisier (1981) and Carl (1985). However, in this paper we present a self-contained proof of this result. Using this theorem we establish that in a bimatrix game with n x n payoff matrices A, B, if the number of non-zero entries in any column of A+B is at most s then an ε-Nash equilibrium of the game can be computed in time nO(log s/ε2}). This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity s. Moreover, for arbitrary bimatrix games---since s can be at most n---the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003). The approximate Carathéodory's theorem also leads to an additive approximation algorithm for the densest k-bipartite subgraph problem. Given a graph with n vertices and maximum degree d, the developed algorithm determines a k x k bipartite subgraph with density within ε (in the additive sense) of the optimal density in time nO(log d/ε2).
{"title":"Approximating Nash Equilibria and Dense Bipartite Subgraphs via an Approximate Version of Caratheodory's Theorem","authors":"Siddharth Barman","doi":"10.1145/2746539.2746566","DOIUrl":"https://doi.org/10.1145/2746539.2746566","url":null,"abstract":"We present algorithmic applications of an approximate version of Caratheodory's theorem. The theorem states that given a set of vectors X in Rd, for every vector in the convex hull of X there exists an ε-close (under the p-norm distance, for 2 ≤ p < ∞) vector that can be expressed as a convex combination of at most b vectors of X, where the bound b depends on ε and the norm p and is independent of the dimension d. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pisier (1981) and Carl (1985). However, in this paper we present a self-contained proof of this result. Using this theorem we establish that in a bimatrix game with n x n payoff matrices A, B, if the number of non-zero entries in any column of A+B is at most s then an ε-Nash equilibrium of the game can be computed in time nO(log s/ε2}). This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity s. Moreover, for arbitrary bimatrix games---since s can be at most n---the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003). The approximate Carathéodory's theorem also leads to an additive approximation algorithm for the densest k-bipartite subgraph problem. Given a graph with n vertices and maximum degree d, the developed algorithm determines a k x k bipartite subgraph with density within ε (in the additive sense) of the optimal density in time nO(log d/ε2).","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79833213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A distance oracle is a compact data structure capable of quickly estimating distances in a given graph. In this paper we provide a new construction for distance oracles in general undirected weighted graphs. Our data structure, for any integer k, requires O( n1+1/k) space, guarantees a stretch of 2k-1, and answers any query in only O(1) time.
{"title":"Approximate Distance Oracles with Improved Bounds","authors":"S. Chechik","doi":"10.1145/2746539.2746562","DOIUrl":"https://doi.org/10.1145/2746539.2746562","url":null,"abstract":"A distance oracle is a compact data structure capable of quickly estimating distances in a given graph. In this paper we provide a new construction for distance oracles in general undirected weighted graphs. Our data structure, for any integer k, requires O( n1+1/k) space, guarantees a stretch of 2k-1, and answers any query in only O(1) time.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74019395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The papers in this volume were presented at the Forty-Seventh Annual ACM Symposium on Theory of Computing (STOC 2015), held as part of the Federated Computing Research Conference in Portland, Oregon, June 15-June 17, 2015. The Symposium was sponsored by the ACM Special Interest Group on Algorithms and Computation Theory (SIGACT). On June 14, the day before STOC, there was a program of workshops and tutorials organized by Chandra Chekuri and Sanjeev Khanna. The workshop was on "Algorithmic Frontiers of Modern Massively Parallel Computation"; the tutorials were on "Hardness and Equivalences for Problems in P" and "Sampling and Volume Computation in High Dimension". In response to a Call for Papers, 347 submissions were received by the submission deadline of November 4, 2014, 3:59PM EST. The Program Committee began its deliberations electronically on December 22, 2014 and continued in that medium until its meeting at MIT in Cambridge, MA on January 30 - February 1, 2015, where final decisions were made. All 26 Program Committee members attended the Program Committee meeting. The Program Committee selected 93 papers for presentation. The submissions were not refereed, and many of these papers represent reports of continuing research. It is expected that most of them will appear in a more polished and complete form in scientific journals. The Program Committee would like to thank all authors who submitted papers for consideration. From among many excellent candidates, the papers "Exponential Separation of Information and Communication for Boolean Function", by Anat Ganor, Gillat Kol and Ran Raz, "2-Server PIR with sub-polynomial communication" by Zeev Dvir and Sivakanth Gopi, and "Lower bounds on the size of semidefinite programming relaxations" by James Lee, Prasad Raghavendra and David Steurer, were selected for the STOC Best Paper Award. The paper "Inapproximability of Nash Equilibrium", by Aviad Rubinstein, was selected for the Danny Lewin Best Student Paper Award.
{"title":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","authors":"R. Servedio, R. Rubinfeld","doi":"10.1145/2746539","DOIUrl":"https://doi.org/10.1145/2746539","url":null,"abstract":"The papers in this volume were presented at the Forty-Seventh Annual ACM Symposium on Theory of Computing (STOC 2015), held as part of the Federated Computing Research Conference in Portland, Oregon, June 15-June 17, 2015. The Symposium was sponsored by the ACM Special Interest Group on Algorithms and Computation Theory (SIGACT). On June 14, the day before STOC, there was a program of workshops and tutorials organized by Chandra Chekuri and Sanjeev Khanna. The workshop was on \"Algorithmic Frontiers of Modern Massively Parallel Computation\"; the tutorials were on \"Hardness and Equivalences for Problems in P\" and \"Sampling and Volume Computation in High Dimension\". \u0000 \u0000In response to a Call for Papers, 347 submissions were received by the submission deadline of November 4, 2014, 3:59PM EST. The Program Committee began its deliberations electronically on December 22, 2014 and continued in that medium until its meeting at MIT in Cambridge, MA on January 30 - February 1, 2015, where final decisions were made. All 26 Program Committee members attended the Program Committee meeting. \u0000 \u0000The Program Committee selected 93 papers for presentation. The submissions were not refereed, and many of these papers represent reports of continuing research. It is expected that most of them will appear in a more polished and complete form in scientific journals. The Program Committee would like to thank all authors who submitted papers for consideration. \u0000 \u0000From among many excellent candidates, the papers \"Exponential Separation of Information and Communication for Boolean Function\", by Anat Ganor, Gillat Kol and Ran Raz, \"2-Server PIR with sub-polynomial communication\" by Zeev Dvir and Sivakanth Gopi, and \"Lower bounds on the size of semidefinite programming relaxations\" by James Lee, Prasad Raghavendra and David Steurer, were selected for the STOC Best Paper Award. The paper \"Inapproximability of Nash Equilibrium\", by Aviad Rubinstein, was selected for the Danny Lewin Best Student Paper Award.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74840793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Primitive coherent configurations (PCCs) are edge-colored digraphs that generalize strongly regular graphs (SRGs), a class perceived as difficult for Graph Isomorphism (GI). Moreover, PCCs arise naturally as obstacles to combinatorial divide-and-conquer approaches for general GI. In a natural sense, the isomorphism problem for PCCs is a stepping stone between SRGs and general GI. In his 1981 paper in the Annals of Math., Babai proposed a combinatorial approach to GI testing via an analysis of the standard individualization/refinement (I/R) technique and proved that I/R yields canonical forms of PCCs in time exp(~O(n1/2)). (The tilde hides polylogarithmic factors.) We improve this bound to exp(~O(n1/3)). This is faster than the current best bound, exp(~O(n1/2)), for general GI, and subsumes Spielman's exp(~O(n1/3)) bound for SRGs (STOC'96, only recently improved to exp(~O(n1/5)) by the present authors and their coauthors (FOCS'13)). Our result implies an exp(~O(n1/3)) upper bound on the number of automorphisms of PCCs with certain easily described and recognized exceptions, making the first progress in 33 years on an old conjecture of Babai. The emergence of exceptions illuminates the technical difficulties: we had to separate these cases from the rest. For the analysis we develop a new combinatorial structure theory for PCCs that in particular demonstrates the presence of "asymptotically uniform clique geometries" among the constituent graphs of PCCs in a certain range of the parameters. A corollary to Babai's 1981 result was an exp(~O(n1/2)) upper bound on the order of primitive but not doubly transitive permutation groups, solving a then 100-year old problem in group theory. An improved bound of exp(~O(n1/3)) (with known exceptions) follows from our combinatorial result. This bound was previously known (Cameron, 1981) only through the Classification of Finite Simple Groups. We note that upper bounds on the order of primitive permutation groups are central to the application of Luks's group theoretic divide-and-conquer methods to GI.
{"title":"Faster Canonical Forms for Primitive Coherent Configurations: Extended Abstract","authors":"Xiaorui Sun, John Wilmes","doi":"10.1145/2746539.2746617","DOIUrl":"https://doi.org/10.1145/2746539.2746617","url":null,"abstract":"Primitive coherent configurations (PCCs) are edge-colored digraphs that generalize strongly regular graphs (SRGs), a class perceived as difficult for Graph Isomorphism (GI). Moreover, PCCs arise naturally as obstacles to combinatorial divide-and-conquer approaches for general GI. In a natural sense, the isomorphism problem for PCCs is a stepping stone between SRGs and general GI. In his 1981 paper in the Annals of Math., Babai proposed a combinatorial approach to GI testing via an analysis of the standard individualization/refinement (I/R) technique and proved that I/R yields canonical forms of PCCs in time exp(~O(n1/2)). (The tilde hides polylogarithmic factors.) We improve this bound to exp(~O(n1/3)). This is faster than the current best bound, exp(~O(n1/2)), for general GI, and subsumes Spielman's exp(~O(n1/3)) bound for SRGs (STOC'96, only recently improved to exp(~O(n1/5)) by the present authors and their coauthors (FOCS'13)). Our result implies an exp(~O(n1/3)) upper bound on the number of automorphisms of PCCs with certain easily described and recognized exceptions, making the first progress in 33 years on an old conjecture of Babai. The emergence of exceptions illuminates the technical difficulties: we had to separate these cases from the rest. For the analysis we develop a new combinatorial structure theory for PCCs that in particular demonstrates the presence of \"asymptotically uniform clique geometries\" among the constituent graphs of PCCs in a certain range of the parameters. A corollary to Babai's 1981 result was an exp(~O(n1/2)) upper bound on the order of primitive but not doubly transitive permutation groups, solving a then 100-year old problem in group theory. An improved bound of exp(~O(n1/3)) (with known exceptions) follows from our combinatorial result. This bound was previously known (Cameron, 1981) only through the Classification of Finite Simple Groups. We note that upper bounds on the order of primitive permutation groups are central to the application of Luks's group theoretic divide-and-conquer methods to GI.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77862971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A randomized encoding allows to express a "complex" computation, given by a function f and input x, by a "simple to compute" randomized representation f(x) whose distribution encodes f(x), while revealing nothing else regarding f and x. Existing randomized encodings, geared mostly to allow encoding with low parallel-complexity, have proven instrumental in various strong applications such as multiparty computation and parallel cryptography. This work focuses on another natural complexity measure: the time required to encode. We construct succinct randomized encodings where the time to encode a computation, given by a program Π and input x, is essentially independent of Π's time complexity, and only depends on its space complexity, as well as the size of its input, output, and description. The scheme guarantees computational privacy of (Π,x), and is based on indistinguishability obfuscation for a relatively simple circuit class, for which there exist instantiations based on polynomial hardness assumptions on multi-linear maps. We then invoke succinct randomized encodings to obtain several strong applications, including: Succinct indistinguishability obfuscation, where the obfuscated program IObf({Π}) computes the same function as Π for inputs x of apriori-bounded size. Obfuscating Π is roughly as fast as encoding the computation of Π on any such input x. Here we also require subexponentially-secure indistinguishability obfuscation for circuits. Succinct functional encryption, where a functional decryption key corresponding to Π allows decrypting Π(x) from encryptions of any plaintext x of apriori-bounded size. Key derivation is as fast as encoding the corresponding computation. Succinct reusable garbling, a stronger form of randomized encodings where any number of inputs x can be encoded separately of Π, independently of Π's time and space complexity. Publicly-verifiable 2-message delegation where verifying the result of a long computation given by Π and input x is as fast as encoding the corresponding computation. We also show how to transform any 2-message delegation scheme to an essentially non-interactive system where the verifier message is reusable. Previously, succinct randomized encodings or any of the above applications were only known based on various non-standard knowledge assumptions. At the heart of our techniques is a generic method of compressing a piecemeal garbled computation, without revealing anything about the secret randomness utilized for garbling.
{"title":"Succinct Randomized Encodings and their Applications","authors":"Nir Bitansky, Sanjam Garg, Sidharth Telang","doi":"10.1145/2746539.2746574","DOIUrl":"https://doi.org/10.1145/2746539.2746574","url":null,"abstract":"A randomized encoding allows to express a \"complex\" computation, given by a function f and input x, by a \"simple to compute\" randomized representation f(x) whose distribution encodes f(x), while revealing nothing else regarding f and x. Existing randomized encodings, geared mostly to allow encoding with low parallel-complexity, have proven instrumental in various strong applications such as multiparty computation and parallel cryptography. This work focuses on another natural complexity measure: the time required to encode. We construct succinct randomized encodings where the time to encode a computation, given by a program Π and input x, is essentially independent of Π's time complexity, and only depends on its space complexity, as well as the size of its input, output, and description. The scheme guarantees computational privacy of (Π,x), and is based on indistinguishability obfuscation for a relatively simple circuit class, for which there exist instantiations based on polynomial hardness assumptions on multi-linear maps. We then invoke succinct randomized encodings to obtain several strong applications, including: Succinct indistinguishability obfuscation, where the obfuscated program IObf({Π}) computes the same function as Π for inputs x of apriori-bounded size. Obfuscating Π is roughly as fast as encoding the computation of Π on any such input x. Here we also require subexponentially-secure indistinguishability obfuscation for circuits. Succinct functional encryption, where a functional decryption key corresponding to Π allows decrypting Π(x) from encryptions of any plaintext x of apriori-bounded size. Key derivation is as fast as encoding the corresponding computation. Succinct reusable garbling, a stronger form of randomized encodings where any number of inputs x can be encoded separately of Π, independently of Π's time and space complexity. Publicly-verifiable 2-message delegation where verifying the result of a long computation given by Π and input x is as fast as encoding the corresponding computation. We also show how to transform any 2-message delegation scheme to an essentially non-interactive system where the verifier message is reusable. Previously, succinct randomized encodings or any of the above applications were only known based on various non-standard knowledge assumptions. At the heart of our techniques is a generic method of compressing a piecemeal garbled computation, without revealing anything about the secret randomness utilized for garbling.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83028031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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