We may enforce an information flow policy by encrypting a protected resource and ensuring that only users authorized by the policy are able to decrypt the resource. In most schemes in the literature that use symmetric cryptographic primitives, each user is assigned a single secret and derives decryption keys using this secret and publicly available information. Recent work has challenged this approach by developing schemes, based on a chain partition of the information flow policy, that do not require public information for key derivation, the trade-off being that a user may need to be assigned more than one secret. In general, many different chain partitions exist for the same policy and, until now, it was not known how to compute an appropriate one. In this paper, we introduce the notion of a tree partition, of which chain partitions are a special case. We show how a tree partition may be used to define a cryptographic enforcement scheme and prove that such schemes can be instantiated in such a way as to preserve the strongest security properties known for cryptographic enforcement schemes. We establish a number of results linking the amount of secret material that needs to be distributed to users with a weighted acyclic graph derived from the tree partition. These results enable us to develop efficient algorithms for deriving tree and chain partitions that minimize the amount of secret material that needs to be distributed.
In an edge-based scheme, public information is associated with each pair @math where @math from which @math can be extracted with knowledge of @math . Thus, informally, we might define @math to be @math , where @math is some symmetric encryption algorithm with key @math contained in @math . An edge-based scheme can be used for arbitrary posets but requires public information @cite_19 .
@cite_19
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For an unweighted graph @math , @math is a subgraph if @math , and @math is a Steiner graph if @math , and for any pair of vertices @math , the distance between them in @math (denoted @math ) is at least the distance between them in @math (denoted @math ). In this paper we introduce the notion of distance preserver. A subgraph (resp., Steiner graph) @math of a graph @math is a subgraph (resp., Steiner) @math -preserver of @math if for every pair of vertices @math with @math , @math . We show that any graph (resp., digraph) has a subgraph @math -preserver with at most @math edges (resp., arcs), and there are graphs and digraphs for which any undirected Steiner @math -preserver contains @math edges. However, we show that if one allows a directed Steiner (diSteiner) @math -preserver, then these bounds can be improved. Specifically, we show that for any graph or digraph there exists a diSteiner @math -preserver with @math arcs, and that this result is tight up to a constant factor. We also study @math -preserving distance labeling schemes, that are labeling schemes that guarantee precise calculation of distances between pairs of vertices that are at a distance of at least @math one from another. We show that there exists a @math -preserving labeling scheme with labels of size @math , and that labels of size @math are required for any @math -preserving labeling scheme. We consider the multiple-unicast problem with three source–terminal pairs over directed acyclic networks with unit-capacity edges. The three @math pairs wish to communicate at unit-rate via network coding. The connectivity between the @math pairs is quantified by means of a connectivity-level vector, @math such that there exist @math edge-disjoint paths between @math and @math . In this paper, we attempt to classify networks based on the connectivity level. It can be observed that unit-rate transmission can be supported by routing if @math , for all @math . In this paper, we consider connectivity-level vectors such that $ i = 1, , 3 k_i . We present either a constructive linear network coding scheme or an instance of a network that cannot support the desired unit-rate requirement, for all such connectivity-level vectors except the vector [1 2 4] (and its permutations). The benefits of our schemes extend to networks with higher and potentially different edge capacities. Specifically, our experimental results indicate that for networks where the different source–terminal paths have a significant overlap, our constructive unit-rate schemes can be packed along with routing to provide higher throughput as compared to a pure routing approach. Let @math be a weighted undirected graph having nonnegative edge weights. An estimate @math of the actual distance @math between @math is said to be of stretch @math if and only if @math . Computing all-pairs small stretch distances efficiently (both in terms of time and space) is a well-studied problem in graph algorithms. We present a simple, novel, and generic scheme for all-pairs approximate shortest paths. Using this scheme and some new ideas and tools, we design faster algorithms for all-pairs @math -stretch distances for a whole range of stretch @math , and we also answer an open question posed by Thorup and Zwick in their seminal paper [J. ACM, 52 (2005), pp. 1-24]. Given a large data matrix @math , we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution @math , or instead @math contains a principal submatrix @math whose entries have marginal distribution @math . As a special case, the hidden (or planted) clique problem requires to find a planted clique in an otherwise uniformly random graph. Assuming unbounded computational resources, this hypothesis testing problem is statistically solvable provided @math for a suitable constant @math . However, despite substantial effort, no polynomial time algorithm is known that succeeds with high probability when @math . Recently Meka and Wigderson meka2013association , proposed a method to establish lower bounds within the Sum of Squares (SOS) semidefinite hierarchy. Here we consider the degree- @math SOS relaxation, and study the construction of meka2013association to prove that SOS fails unless @math . An argument presented by Barak implies that this lower bound cannot be substantially improved unless the witness construction is changed in the proof. Our proof uses the moments method to bound the spectrum of a certain random association scheme, i.e. a symmetric random matrix whose rows and columns are indexed by the edges of an Erd "os-Renyi random graph. A graph @math is called @math -full if @math contains a subgraph @math of @math edges such that, for any non-empty @math , @math holds. Here, @math denotes the set of vertices incident to @math . It is known that the family of edge sets of @math -full graphs forms a family of matroid, known as the sparsity matroid of @math . In this paper, we give a constant-time approximation algorithm for the rank of the sparsity matroid of a degree-bounded undirected graph. This leads to a constant-time tester for @math -fullness in the bounded-degree model, (i.e., we can decide with high probability whether an input graph satisfies a property @math or far from @math ). Depending on the values of @math and @math , it can test various properties of a graph such as connectivity, rigidity, and how many spanning trees can be packed. Based on this result, we also propose a constant-time tester for @math -edge-connected-orientability in the bounded-degree model, where an undirected graph @math is called @math -edge-connected-orientable if there exists an orientation @math of @math with a vertex @math such that @math contains @math arc-disjoint dipaths from @math to each vertex @math and @math arc-disjoint dipaths from each vertex @math to @math . A tester is called a one-sided error tester for @math if it always accepts a graph satisfying @math . We show, for @math and (proper) @math , any one-sided error tester for @math -fullness and @math -edge-connected-orientability requires @math queries. For a positive parameter @math , the @math -bounded distance between a pair of vertices @math in a weighted undirected graph @math is the length of the shortest @math path in @math with at most @math edges, aka hops . For @math as above and @math , a @math -hopset of @math is a graph @math on the same vertex set, such that all distances in @math are @math -approximated by @math -bounded distances in @math . Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with @math edges, or with a hopbound @math . In this paper we devise a construction of linear-size hopsets with hopbound @math . This improves the previous bound almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm EN16 for computing hopsets with a constant (i.e., independent of @math ) hopbound requires @math time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from EN16 . We use our hopsets to devise a distributed routing scheme that exhibits near-optimal tradeoff between individual memory requirement @math of vertices throughout preprocessing and routing phases of the algorithm, and stretch @math , along with a near-optimal construction time @math , where @math is the hop-diameter of the input graph. We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property @math is called a @math -graph . If @math satisfies certain properties, then an n-node m-edge @math -graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most @math bits for any continuous superadditive function @math so that there are at most @math distinct @math -node @math -graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such @math include all conjunctions over the following groups of properties: (1) G is a planar graph or a plane graph; (2) @math is directed or undirected; (3) @math is triangulated, triconnected, biconnected, merely connected, or not required to be connected; (4) the nodes of G are labeled with labels from @math for @math ; (5) the edges of G are labeled with labels from @math for @math ; and (6) each node (respectively, edge) of G has at most @math self-loops (respectively, @math multiple edges). Moreover, @math and @math are not required to be O(1) for the cases of @math being a plane triangulation. These examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs, other than rooted trees, with known polynomial-time information-theoretically optimal coding schemes. This paper presents a family of memory-balanced routing schemes that use relatively short paths while storing relatively little routing information. The quality of the routes provided by a scheme is measured in terms of their stretch, namely, the maximum ratio between the length of a route connecting some pair of processors and their distance. The hierarchical schemes @math (for every integer @math ) presented in this paper guarantee a stretch factor of @math on the length of the routes and require storing at most @math bits of routing information per vertex in an n-processor network with diameter D. The schemes are name independent and applicable to general networks with arbitrary edge weights. This improves on previous designs whose stretch bound was exponential in k.The proposed schemes are based on a new efficient solution to a certain graph-theoretic problem concerning sparse graph covers. The new cover technique has already found several other a... We study undirected networks with edge costs that satisfy the triangle inequality. Let @math denote the number of nodes. We present an @math -approximation algorithm for a generalization of the metric-cost subset @math -node-connectivity problem. Our approximation guarantee is proved via lower bounds that apply to the simple edge-connectivity version of the problem, where the requirements are for edge-disjoint paths rather than for openly node-disjoint paths. A corollary is that, for metric costs and for each @math , there exists a @math -node connected graph whose cost is within a factor of @math of the cost of any simple @math -edge connected graph. Based on our @math -approximation algorithm, we present an @math -approximation algorithm for the metric-cost node-connectivity survivable network design problem, where @math denotes the maximum requirement over all pairs of nodes. Our results contrast with the case of edge costs of 0 or 1, where Kortsarz, Krauthgamer, and Lee. [SIAM J. Comput., 33 (2004), pp. 704-720] recently proved, assuming NP @math DTIME( @math ), a hardness-of-approximation lower bound of @math for the subset @math -node-connectivity problem, where @math denotes a small positive number. This paper investigates a zero-sum game played on a weighted connected graph @math between two players, the tree player and the edge player. At each play, the tree player chooses a spanning tree @math and the edge player chooses an edge @math . The payoff to the edge player is @math , defined as follows: If @math lies in the tree @math then @math ; if @math does not lie in the tree then @math , where @math is the weight of edge @math and @math is the weight of the unique cycle formed when edge @math is added to the tree @math . The main result is that the value of the game on any @math -vertex graph is bounded above by @math . It is conjectured that the value of the game is @math . The game arises in connection with the @math -server problem on a road network; i.e., a metric space that can be represented as a multigraph @math in which each edge @math represents a road of length @math . It is shown that, if the value of the game on @math is @math , then there is a randomized strategy that achieves a competitive ratio of @math against any oblivious adversary. Thus, on any @math -vertex road network, there is a randomized algorithm for the @math -server problem that is @math competitive against oblivious adversaries. At the heart of the analysis of the game is an algorithm that provides an approximate solution for the simple network design problem. Specifically, for any @math -vertex weighted, connected multigraph, the algorithm constructs a spanning tree @math such that the average, over all edges @math , of @math is less than or equal to @math . This result has potential application to the design of communication networks. It also improves substantially known estimates concerning the existence of a sparse basis for the cycle space of a graph. We show that every weighted connected graph @math contains as a subgraph a spanning tree into which the edges of @math can be embedded with average stretch @math . Moreover, we show that this tree can be constructed in time @math in general, and in time @math if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for symmetric diagonally dominant linear systems of Spielman and Teng from @math to @math , and to @math when the system is planar. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees. In the subgraph-freeness problem, we are given a constant-size graph @math , and wish to determine whether the network contains @math as a subgraph or not. The relaxation of the problem only requires us to distinguish graphs that are @math -free from graphs that are @math -far from @math -free, meaning an @math -fraction of their edges must be removed to obtain an @math -free graph. Recently, Censor-Hillel et. al. and showed that in the property-testing regime it is possible to test @math -freeness for any graph @math of size 4 in constant time, @math rounds, regardless of the network size. However, Fraigniaud et. al. also showed that their techniques for graphs @math of size 4 cannot test @math -cycle-freeness in constant time. In this paper we revisit the subgraph-freeness problem and show that @math -cycle-freeness, and indeed @math -freeness for many other graphs @math comprising more than 4 vertices, can be tested in constant time. We show that @math -freeness can be tested in @math rounds for any cycle @math , improving on the running time of @math of the previous algorithms for triangle-freeness and @math -freeness. In the special case of triangles, we show that triangle-freeness can be solved in @math rounds independently of @math , when @math is not too small with respect to the number of nodes and edges. We also show that @math -freeness for any constant-size tree @math can be tested in @math rounds, even without the property-testing relaxation. Building on these results, we define a general class of graphs for which we can test subgraph-freeness in @math rounds. This class includes all graphs over 5 vertices except the 5-clique, @math . For cliques @math over @math nodes, we show that @math -freeness can be tested in @math rounds, where @math is the number of edges. The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers @math and @math , the minimum size @math of the edge boundary of an @math -element subset of @math ; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on @math . We show that for any @math -element subset @math and any integer @math , if the edge boundary of @math has size at most @math , then there exists an extremal family @math such that @math , where @math is an absolute constant. This is best-possible, up to the value of @math . Our result can be seen as a stability' version of the edge isoperimetric inequality in the discrete cube, and as a discrete analogue of the seminal stability result of Fusco, Maggi and Pratelli concerning the isoperimetric inequality in Euclidean space. In this work, we study indistinguishability obfuscation and functional encryption for general circuits: Indistinguishability obfuscation requires that given any two equivalent circuits @math and @math of similar size, the obfuscations of @math and @math should be computationally indistinguishable. In functional encryption, ciphertexts encrypt inputs @math and keys are issued for circuits @math . Using the key @math to decrypt a ciphertext @math yields the value @math but does not reveal anything else about @math . Furthermore, no collusion of secret key holders should be able to learn anything more than the union of what they can each learn individually. We give constructions for indistinguishability obfuscation and functional encryption that supports all polynomial-size circuits. We accomplish this goal in three steps: (1) We describe a candidate construction for indistinguishability obfuscation for @math circuits. The security of this construction is based on a new al... Given an undirected graph @math , it is known that its edge-connectivity @math can be computed by solving @math max-flow problems. The best time bounds known for the problem are @math , due to Matula (28th IEEE Symposium on the Foundations of Computer Science, 1987, pp. 249–251) if G is simple, and @math , due to Even and Tarjan (SIAM J. Comput., 4 (1975), pp. 507–518) if G is multiple.An @math time algorithm for computing the edge-connectivity @math of a multigraph @math , where @math is the number of pairs of nodes between which G has an edge, is proposed. This algorithm does not use any max-flow algorithm but consists only of @math times of graph searches and edge contractions. This method is then extended to a capacitated network to compute its minimum cut capacity in @math time. The focus of this paper is on the public communication required for generating a maximal-rate secret key (SK) within the multiterminal source model of Csiszar and Narayan. Building on the prior work of Tyagi for the two-terminal scenario, we derive a lower bound on the communication complexity, @math , defined to be the minimum rate of public communication needed to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by @math , is an upper bound on @math . For the class of pairwise independent network (PIN) models defined on uniform hypergraphs, we show that a certain Type @math condition, which is verifiable in polynomial time, guarantees that our lower bound on @math meets the @math upper bound. Thus, the PIN models satisfying our condition are @math -maximal, indicating that the upper bound @math holds with equality. This allows us to explicitly evaluate @math for such PIN models. We also give several examples of PIN models that satisfy our Type @math condition. Finally, we prove that for an arbitrary multiterminal source model, a stricter version of our Type @math condition implies that communication from all terminals (omnivocality) is needed for establishing an SK of maximum rate. For three-terminal source models, the converse is also true: omnivocality is needed for generating a maximal-rate SK only if the strict Type @math condition is satisfied. However, for the source models with four or more terminals, counterexamples exist showing that the converse does not hold in general. In the three-dimensional (3D) strip packing problem, we are given a set of 3D rectangular items and a 3D box @math . The goal is to pack all the items in @math such that the height of the packing is minimized. We consider the most basic version of the problem, where the items must be packed with their edges parallel to the edges of @math and cannot be rotated. Building upon Caprara's work for the two-dimensional (2D) bin packing problem, we obtain an algorithm that, given any @math , achieves an approximation of @math , where @math is the well-known number that occurs naturally in the context of bin packing. Our key idea is to establish a connection between bin packing solutions for an arbitrary instance @math and the strip packing solutions for the corresponding instance obtained from @math by applying the harmonic transformation to certain dimensions. Based on this connection, we also give a simple alternate proof of the @math approximation for 2D ... In the generalized Russian cards problem, the three players Alice, Bob and Cath draw @math a , b and @math c cards, respectively, from a deck of @math a + b + c cards. Players only know their own cards and what the deck of cards is. Alice and Bob are then required to communicate their hand of cards to each other by way of public messages. For a natural number @math k , the communication is said to be @math k -safe if Cath does not learn whether or not Alice holds any given set of at most @math k cards that are not Cath's, a notion originally introduced as weak @math k -security by Swanson and Stinson. An elegant solution by Atkinson views the cards as points in a finite projective plane. We propose a general solution in the spirit of Atkinson's, although based on finite vector spaces rather than projective planes, and call it the geometric protocol'. Given arbitrary @math c , k > 0 , this protocol gives an informative and @math k -safe solution to the generalized Russian cards problem for infinitely many values of @math ( a , b , c ) with @math b = O ( a c ) . This improves on the collection of parameters for which solutions are known. In particular, it is the first solution which guarantees @math k -safety when Cath has more than one card. The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in @math time, where @math is the number of vertices in the input graph @math . Peleg and Rubinovich (FOCS'99) showed a lower bound of @math for this problem, where @math is the hop-diameter of @math . Whether or not this problem can be solved in @math time when @math is relatively small is a major notorious open question. Despite intensive research LP13,N14,HKN15,EN16,BKKL16 that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this paper we answer this question in the affirmative. We devise an algorithm that requires @math time, for @math , and @math time, for larger @math . This running time is sublinear in @math in almost the entire range of parameters, specifically, for @math . For the all-pairs shortest paths problem, our algorithm requires @math time, regardless of the value of @math . We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in the multipass semi-streaming model of computation. From the technical viewpoint, our algorithm computes a hopset @math of a skeleton graph @math of @math without first computing @math itself. We then conduct a Bellman-Ford exploration in @math , while computing the required edges of @math on the fly. As a result, our algorithm computes exactly those edges of @math that it really needs, rather than computing approximately the entire @math . Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090.