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.
The significant open problem with prior work on chain-based schemes is the assumption that the chain partition is part of the input to the algorithm: there may be many such partitions and it is not immediately obvious how one should select a specific partition in order to optimize characteristics of the corresponding enforcement scheme (an example being to minimize the number of secrets issued). Hence, it seems very natural to ask how difficult it is to compute a good'' chain partition, given that Our recent work @cite_2 shows that it is possible to compute a minimal chain partition in polynomial time using a minimum cost network flow algorithm.
@cite_2
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The simple security property in an information flow policy can be enforced by encrypting data objects and distributing an appropriate secret to each user. A user derives a suitable decryption key from the secret and publicly available information. A chain-based enforcement scheme provides an alternative method of cryptographic enforcement that does not require any public information, the trade-off being that a user may require more than one secret. For a given information flow policy, there will be many different possible chain-based enforcement schemes. In this paper, we provide a polynomial-time algorithm for selecting a chain-based scheme which uses the minimum possible number of secrets. We also compute the number of secrets that will be required and establish an upper bound on the number of secrets required by any user. Partitioning of circuit netlists in VLSI design is considered. It is shown that the second smallest eigenvalue of a matrix derived from the netlist gives a provably good approximation of the optimal ratio cut partition cost. It is also demonstrated that fast Lanczos-type methods for the sparse symmetric eigenvalue problem are a robust basis for computing heuristic ratio cuts based on the eigenvector of this second eigenvalue. Effective clustering methods are an immediate by-product of the second eigenvector computation and are very successful on the difficult input classes proposed in the CAD literature. The intersection graph representation of the circuit netlist is considered, as a basis for partitioning, a heuristic based on spectral ratio cut partitioning of the netlist intersection graph is proposed. The partitioning heuristics were tested on industry benchmark suites, and the results were good in terms of both solution quality and runtime. Several types of algorithmic speedups and directions for future work are discussed. > We present a novel approach to proving the absence of timing channels. The idea is to partition the programâ s execution traces in such a way that each partition component is checked for timing attack resilience by a time complexity analysis and that per-component resilience implies the resilience of the whole program. We construct a partition by splitting the program traces at secret-independent branches. This ensures that any pair of traces with the same public input has a component containing both traces. Crucially, the per-component checks can be normal safety properties expressed in terms of a single execution. Our approach is thus in contrast to prior approaches, such as self-composition, that aim to reason about multiple (kâ � 2) executions at once. We formalize the above as an approach called quotient partitioning, generalized to any k-safety property, and prove it to be sound. A key feature of our approach is a demand-driven partitioning strategy that uses a regex-like notion called trails to identify sets of execution traces, particularly those influenced by tainted (or secret) data. We have applied our technique in a prototype implementation tool called Blazer, based on WALA, PPL, and the brics automaton library. We have proved timing-channel freedom of (or synthesized an attack specification for) 24 programs written in Java bytecode, including 6 classic examples from the literature and 6 examples extracted from the DARPA STAC challenge problems. The present paper introduces and studies a new combinatorial clustering model for the consolidation of farmland. While the general problem turns out to be NP-hard even in quite restricted cases, the Size-restricted Minimum-k-Star Group Partition problem is solvable in polynomial time. Based on this tractability result, we derive a general approximation algorithm which, as the mathematical analysis and economic evaluation shows, performs well in theory and practice. Y. Bartal (1996, 1998) gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O (log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his result are inherently randomized. In this paper we derandomize the use of Bartal's algorithm in the design of approximation algorithms. We give an efficient polynomial time algorithm that given a finite n point metric G, constructs O(n log n) trees and a probability distribution spl mu on them such that the expected stretch of any edge of G in a tree chosen according to spl mu is at most O(log n log log n). Our result establishes that finite metrics can be probabilistically approximated by a small number of tree metrics. We obtain the first deterministic approximation algorithms for buy-at-bulk network design and vehicle routing; in addition we subsume results from our earlier work on derandomization. Our main result is obtained by a novel view of probabilistic approximation of metric spaces as a deterministic optimization problem via linear programming. In this paper we present a parallel formulation of the multilevel graph partitioning and sparse matrix ordering algorithm. A key feature of our parallel formulation (that distinguishes it from other proposed parallel formulations of multilevel algorithms) is that it partitions the vertices of the graph intopparts while distributing the overall adjacency matrix of the graph among allpprocessors. This mapping results in substantially smaller communication than one-dimensional distribution for graphs with relatively high degree, especially if the graph is randomly distributed among the processors. We also present a parallel algorithm for computing a minimal cover of a bipartite graph which is a key operation for obtaining a small vertex separator that is useful for computing the fill reducing ordering of sparse matrices. Our parallel algorithm achieves a speedup of up to 56 on 128 processors for moderate size problems, further reducing the already moderate serial run time of multilevel schemes. Furthermore, the quality of the produced partitions and orderings are comparable to those produced by the serial multilevel algorithm that has been shown to outperform both spectral partitioning and multiple minimum degree. This paper introduces the concept of low-congestion shortcuts for (near-)planar networks, and demonstrates their power by using them to obtain near-optimal distributed algorithms for problems such as Minimum Spanning Tree (MST) or Minimum Cut, in planar networks. Consider a graph G = (V, E) and a partitioning of V into subsets of nodes S1, . . ., SN, each inducing a connected subgraph G[Si]. We define an α-congestion shortcut with dilation β to be a set of subgraphs H1, . . ., HN ⊆ G, one for each subset Si, such that 1. For each i ∈ [1, N], the diameter of the subgraph G[Si] + Hi is at most β. 2. For each edge e ∈ E, the number of subgraphs G[Si] + Hi containing e is at most α. We prove that any partition of a D-diameter planar graph into individually-connected parts admits an O(D log D)-congestion shortcut with dilation O(D log D), and we also present a distributed construction of it in O(D) rounds. We moreover prove these parameters to be near-optimal; i.e., there are instances in which, unavoidably, max α, β = Ω(D[EQUATION]). Finally, we use low-congestion shortcuts, and their efficient distributed construction, to derive O(D)-round distributed algorithms for MST and Min-Cut, in planar networks. This complexity nearly matches the trivial lower bound of Ω(D). We remark that this is the first result bypassing the well-known Ω(D + [EQUATION]) existential lower bound of general graphs (see Peleg and Rubinovich [FOCS'99]; Elkin [STOC'04]; and Das [STOC'11]) in a family of graphs of interest. We consider the problem of identifying underlying community-like structures in graphs. Toward this end, we study the stochastic block model (SBM) on k-clusters: a random model on n = km vertices, partitioned in k equal sized clusters, with edges sampled independently across clusters with probability q and within clusters with probability p, p > q. The goal is to recover the initial “hidden” partition of [n]. We study semidefinite programming (SDP)-based algorithms in this context. In the regime (p = (m) m ) and (q = (m) m ), we show that a certain natural SDP-based algorithm solves the problem of exact recovery in the k-community SBM, with high probability, whenever ( - > 1 ), as long as (k=o( n) ). This threshold is known to be the information theoretically optimal. We also study the case when (k= ( (n)) ). In this case however, we achieve recovery guarantees that no longer match the optimal condition ( - > 1 ), thus leaving achieving optimality for this range an open question. We analyze a generalized index coding problem that allows multiple users to request the same packet. For this problem we introduce a novel coding scheme called partition multicast. Our scheme can be seen as a natural generalization of clique cover for directed index coding problems. Further, partition multicast corresponds to an achievable scheme for the generalized bipartite index coding problem that we introduce in this paper. Our scheme partitions the nodes into groups and solves a multicasting problem within each group. We show that Partition Multicast is optimal for a few families of graphs and generalizes previous achievable schemes, namely directed cycle covers. We also show that finding the best partition is computationally intractable to compute in general. In this paper we study the prevalent problem of graph partitioning by analyzing the diffusion-based partitioning heuristic Bubble-FOS C, a key component of a practical successful graph partitioner ( in J. Parallel Distrib. Comput. 69(9):750---761, 2009). We begin by studying the disturbed diffusion scheme FOS C, which computes the similarity measure used in Bubble-FOS C and is therefore the most crucial component. By relating FOS C to random walks, we obtain precise characterizations of the behavior of FOS C on tori and hypercubes. Besides leading to new knowledge on FOS C (and therefore also on Bubble-FOS C), these characterizations have been recently used for the analysis of load balancing algorithms ( in Proceedings of the 22nd Annual Symposium on Discrete Algorithms, pp. 429---439, 2011). We then regard Bubble-FOS C, which has been shown in previous experiments to produce solutions with good partition shapes and other favorable properties. In this paper we prove that it computes a relaxed solution to an edge cut minimizing binary quadratic program (BQP). This result provides the first substantial theoretical insight why Bubble-FOS C yields good experimental results in terms of graph partitioning metrics. Moreover, we show that in bisections computed by Bubble-FOS C, at least one of the two parts is connected. Using the aforementioned relation between FOS C and random walks, we prove that in vertex-transitive graphs both parts must be connected components. Abstract A partition ( C 1 , C 2 , … , C q ) of G = ( V , E ) into clusters of strong (respectively, weak) diameter d , such that the supergraph obtained by contracting each C i is l -colorable is called a strong (resp., weak) ( d , l ) -network-decomposition. Network-decompositions were introduced in a seminal paper by Awerbuch, Goldberg, Luby and Plotkin in 1989. showed that strong ( d , l ) -network-decompositions with d = l = exp ⁡ O ( log ⁡ n log ⁡ log ⁡ n ) can be computed in distributed deterministic time O ( d ) . Even more importantly, they demonstrated that network-decompositions can be used for a great variety of applications in the message-passing model of distributed computing. The result of was improved by Panconesi and Srinivasan in 1992: in the latter result d = l = exp ⁡ O ( log ⁡ n ) , and the running time is O ( d ) as well. In another remarkable breakthrough Linial and Saks (in 1992) showed that weak ( O ( log ⁡ n ) , O ( log ⁡ n ) ) -network-decompositions can be computed in distributed randomized time O ( log 2 ⁡ n ) . Much more recently Barenboim (2012) devised a distributed randomized constant-time algorithm for computing strong network decompositions with d = O ( 1 ) . However, the parameter l in his result is O ( n 1 2 + ϵ ) . In this paper we drastically improve the result of Barenboim and devise a distributed randomized constant-time algorithm for computing strong ( O ( 1 ) , O ( n ϵ ) ) -network-decompositions. As a corollary we derive a constant-time randomized O ( n ϵ ) -approximation algorithm for the distributed minimum coloring problem, improving the previously best-known O ( n 1 2 + ϵ ) approximation guarantee. We also derive other improved distributed algorithms for a variety of problems. Most notably, for the extremely well-studied distributed minimum dominating set problem currently there is no known deterministic polylogarithmic-time algorithm. We devise a deterministic polylogarithmic-time approximation algorithm for this problem, addressing an open problem of Lenzen and Wattenhofer (2010). For a partition of an n -point set (X ) into k subsets (clusters) S 1 ,S 2 ,. . .,S k , we consider the cost function ( i=1 ^k x S_i |x-c(S_i) |^2 ) , where c(S i ) denotes the center of gravity of S i . For k=2 and for any fixed d and e >0 , we present a deterministic algorithm that finds a 2-clustering with cost no worse than (1+e) -times the minimum cost in time O(n log n); the constant of proportionality depends polynomially on e . For an arbitrary fixed k , we get an O(n log k n) algorithm for a fixed e , again with a polynomial dependence on e . We consider the problem of threshold secret sharing in groups with hierarchical structure. In such settings, the secret is shared among a group of participants that is partitioned into levels. The access structure is then determined by a sequence of threshold requirements: a subset of participants is authorized if it has at least k0 0 members from the highest level, as well as at least k1 > k0 members from the two highest levels and so forth. Such problems may occur in settings where the participants differ in their authority or level of confidence and the presence of higher level participants is imperative to allow the recovery of the common secret. Even though secret sharing in hierarchical groups has been studied extensively in the past, none of the existing solutions addresses the simple setting where, say, a bank transfer should be signed by three employees, at least one of whom must be a department manager. We present a perfect secret sharing scheme for this problem that, unlike most secret sharing schemes that are suitable for hierarchical structures, is ideal. As in Shamir's scheme, the secret is represented as the free coefficient of some polynomial. The novelty of our scheme is the usage of polynomial derivatives in order to generate lesser shares for participants of lower levels. Consequently, our scheme uses Birkhoff interpolation, i.e., the construction of a polynomial according to an unstructured set of point and derivative values. A substantial part of our discussion is dedicated to the question of how to assign identities to the participants from the underlying finite field so that the resulting Birkhoff interpolation problem will be well posed. In addition, we devise an ideal and efficient secret sharing scheme for the closely related hierarchical threshold access structures that were studied by Simmons and Brickell. Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc. of the 6th SIAM Conference on Parallel Processing for Scientific Computing, 1993, 445--452; Hendrickson and Leland, A Multilevel Algorithm for Partitioning Graphs, Tech. report SAND 93-1301, Sandia National Laboratories, Albuquerque, NM, 1993]. From the early work it was clear that multilevel techniques held great promise; however, it was not known if they can be made to consistently produce high quality partitions for graphs arising in a wide range of application domains. We investigate the effectiveness of many different choices for all three phases: coarsening, partition of the coarsest graph, and refinement. In particular, we present a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of the size of the final partition obtained after multilevel refinement. We also present a much faster variation of the Kernighan--Lin (KL) algorithm for refining during uncoarsening. We test our scheme on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that our scheme produces partitions that are consistently better than those produced by spectral partitioning schemes in substantially smaller time. Also, when our scheme is used to compute fill-reducing orderings for sparse matrices, it produces orderings that have substantially smaller fill than the widely used multiple minimum degree algorithm. Given a set S of radio stations located on a line and an integer h (1 ≤ h ≤ |S| - 1), the Min Assignment problem is to find a range assignment of minimum power consumption provided that any pair of stations can communicate in at most h hops. Previous positive results for this problem were known only when h = |S| - 1 (i.e. the unbounded case) or when the stations are equally spaced (i.e. the uniform chain). In particular, Kirousis, Kranakis, Krizanc and Pelc (1997) provided an efficient exact solution for the unbounded case and efficient approximated solutions for the uniform chain, respectively. This paper presents the first polynomial time, approximation algorithm for the Min Assignment problem. The algorithm guarantees an approximation ratio of 2 and runs in time O(hn3). We also prove that, for constant h and for "well spread" instances (a broad generalization of the uniform chain case), we can find a solution in time O(hn3) whose cost is at most an (1 + Ɛ(n)) factor from the optimum, where Ɛ(n) = o(1) and n is the number of stations. This result significantly improves the approximability result by on uniform chains. Both of our approximation results are obtained by new algorithms that exactly solves two natural variants of the Min Assignment problem that might have independent interest: the All-To-One problem (in which every station must reach a fixed one in at most h hops) and the Base Location problem (in which the goal is to select a set of Basis among the stations and all the other stations must reach one of them in at most h -1 hops). Finally, we show that for h = 2 the Min Assignment problem can be solved in O(n3)-time. We present a novel approach to fully homomorphic encryption (FHE) that dramatically improves performance and bases security on weaker assumptions. A central conceptual contribution in our work is a new way of constructing leveled fully homomorphic encryption schemes (capable of evaluating arbitrary polynomial-size circuits), without Gentry's bootstrapping procedure. Specifically, we offer a choice of FHE schemes based on the learning with error (LWE) or ring-LWE (RLWE) problems that have 2λ security against known attacks. For RLWE, we have: • A leveled FHE scheme that can evaluate L-level arithmetic circuits with O(λ · L3) per-gate computation -- i.e., computation quasi-linear in the security parameter. Security is based on RLWE for an approximation factor exponential in L. This construction does not use the bootstrapping procedure. • A leveled FHE scheme that uses bootstrapping as an optimization, where the per-gate computation (which includes the bootstrapping procedure) is O(λ2), independent of L. Security is based on the hardness of RLWE for quasi-polynomial factors (as opposed to the sub-exponential factors needed in previous schemes). We obtain similar results to the above for LWE, but with worse performance. Based on the Ring LWE assumption, we introduce a number of further optimizations to our schemes. As an example, for circuits of large width -- e.g., where a constant fraction of levels have width at least λ -- we can reduce the per-gate computation of the bootstrapped version to O(λ), independent of L, by batching the bootstrapping operation. Previous FHE schemes all required Ω(λ3.5) computation per gate. At the core of our construction is a much more effective approach for managing the noise level of lattice-based ciphertexts as homomorphic operations are performed, using some new techniques recently introduced by Brakerski and Vaikuntanathan (FOCS 2011). We study a family of closely-related distributed graph problems, which we call degree splitting, where roughly speaking the objective is to partition (or orient) the edges such that each node's degree is split almost uniformly. Our findings lead to answers for a number of problems, a sampling of which includes: • We present a poly log n round deterministic algorithm for (2Δ−1)·(1+o(1))-edge-coloring, where Δ denotes the maximum degree. Modulo the 1 + o(1) factor, this settles one of the long-standing open problems of the area from the 1990's (see e.g. Panconesi and Srinivasan [PODC'92]). Indeed, a weaker requirement of (2Δ − 1) · poly log Δ-edge-coloring in poly log n rounds was asked for in the 4th open question in the Distributed Graph Coloring book by Barenboim and Elkin. • We show that sinkless orientation---i.e., orienting edges such that each node has at least one out-going edge---on Δ-regular graphs can be solved in O(logΔ log n) rounds randomized and in O(logΔ n) rounds deterministically. These prove the corresponding lower bounds by [STOC'16] and Chang, Kopelowitz, and Pettie [FOCS'16] to be tight. Moreover, these show that sinkless orientation exhibits an exponential separation between its randomized and deterministic complexities, akin to the results of for Δ-coloring Δ-regular trees. • We present a randomized O(log4 n) round algorithm for orienting a-arboricity graphs with maximum out-degree a(1 + e). This can be also turned into a decomposition into a(1 + e) forests when a = Ω(log n) and into a(1 + e) pseduo-forests when a = o(log n). Obtaining an efficient distributed decomposition into less than 2a forests was stated as the 10th open problem in the book by Barenboim and Elkin. Abstract In this paper we study the Steiner tree problem in graphs for the case when vertices as well as edges have weights associated with them. A greedy approximation algorithm based on “spider decompositions” was developed by Klein and Ravi for this problem. This algorithm provides a worst case approximation ratio of 2 ln k , where k is the number of terminals. However, the best known lower bound on the approximation ratio is (1− o (1)) ln k , assuming that NP ⊈ DTIME [ n O (log log n ) ], by a reduction from set cover. We show that for the unweighted case we can obtain an approximation factor of ln k . For the weighted case we develop a new decomposition theorem and generalize the notion of “spiders” to “branch-spiders” that are used to design a new algorithm with a worst case approximation factor of 1.5 ln k . We then generalize the method to yield an approximation factor of (1.35+ e ) ln k , for any constant e >0. These algorithms, although polynomial, are not very practical due to their high running time, since we need to repeatedly find many minimum weight matchings in each iteration. We also develop a simple greedy algorithm that is practical and has a worst case approximation factor of 1.6103 ln k . The techniques developed for this algorithm imply a method of approximating node weighted network design problems defined by 0–1 proper functions as well. These new ideas also lead to improved approximation guarantees for the problem of finding a minimum node weighted connected dominating set. The previous best approximation guarantee for this problem was 3 ln n by Guha and Khuller. By a direct application of the methods developed in this paper we are able to develop an algorithm with an approximation factor of (1.35+ e ) ln n for any fixed e >0. To improve the accuracy of learning result, in practice multiple parties may collaborate through conducting joint Back-Propagation neural network learning on the union of their respective data sets. During this process no party wants to disclose her his private data to others. Existing schemes supporting this kind of collaborative learning are either limited in the way of data partition or just consider two parties. There lacks a solution that allows two or more parties, each with an arbitrarily partitioned data set, to collaboratively conduct the learning. This paper solves this open problem by utilizing the power of cloud computing. In our proposed scheme, each party encrypts his her private data locally and uploads the ciphertexts into the cloud. The cloud then executes most of the operations pertaining to the learning algorithms over ciphertexts without knowing the original private data. By securely offloading the expensive operations to the cloud, we keep the computation and communication costs on each party minimal and independent to the number of participants. To support flexible operations over ciphertexts, we adopt and tailor the BGN "doubly homomorphic" encryption algorithm for the multiparty setting. Numerical analysis and experiments on commodity cloud show that our scheme is secure, efficient, and accurate. We present a multi-level graph partitioning algorithm using novel local improvement algorithms and global search strategies transferred from multigrid linear solvers. Local improvement algorithms are based on max-flow min-cut computations and more localized FM searches. By combining these techniques, we obtain an algorithm that is fast on the one hand and on the other hand is able to improve the best known partitioning results for many inputs. For example, in Walshaw's well known benchmark tables we achieve 317 improvements for the tables at 1 , 3 and 5 imbalance. Moreover, in 118 out of the 295 remaining cases we have been able to reproduce the best cut in this benchmark.