Reaching Approximate Byzantine Consensus with Multi-hop Communication

We address the problem of reaching consensus in the presence of Byzantine faults. In particular, we are interested in investigating the impact of messages relay on the network connectivity for a correct iterative approximate Byzantine consensus algorithm to exist. The network is modeled by a simple directed graph. We assume a node can send messages to another node that is up to $l$ hops away via forwarding by the intermediate nodes on the routes, where $l\in \mathbb{N}$ is a natural number. We characterize the necessary and sufficient topological conditions on the network structure. The tight conditions we found are consistent with the tight conditions identified for $l=1$, where only local communication is allowed, and are strictly weaker for $l>1$. Let $l^*$ denote the length of a longest path in the given network. For $l\ge l^*$ and undirected graphs, our conditions hold if and only if $n\ge 3f+1$ and the node-connectivity of the given graph is at least $2f+1$ , where $n$ is the total number of nodes and $f$ is the maximal number of Byzantine nodes; and for $l\ge l^*$ and directed graphs, our conditions is equivalent to the tight condition found for exact Byzantine consensus. Our sufficiency is shown by constructing a correct algorithm, wherein the trim function is constructed based on investigating a newly introduced minimal messages cover property. The trim function proposed also works over multi-graphs.Comment: 24 pages, 1 figure. arXiv admin note: text overlap with arXiv:1203.188

An Optimal Self-Stabilizing Firing Squad

Consider a fully connected network where up to $t$ processes may crash, and all processes start in an arbitrary memory state. The self-stabilizing firing squad problem consists of eventually guaranteeing simultaneous response to an external input. This is modeled by requiring that the non-crashed processes "fire" simultaneously if some correct process received an external "GO" input, and that they only fire as a response to some process receiving such an input. This paper presents FireAlg, the first self-stabilizing firing squad algorithm. The FireAlg algorithm is optimal in two respects: (a) Once the algorithm is in a safe state, it fires in response to a GO input as fast as any other algorithm does, and (b) Starting from an arbitrary state, it converges to a safe state as fast as any other algorithm does.Comment: Shorter version to appear in SSS0

Assignment Problems of Different-Sized Inputs in MapReduce

A MapReduce algorithm can be described by a mapping schema, which assigns inputs to a set of reducers, such that for each required output there exists a reducer that receives all the inputs that participate in the computation of this output. Reducers have a capacity, which limits the sets of inputs that they can be assigned. However, individual inputs may vary in terms of size. We consider, for the first time, mapping schemas where input sizes are part of the considerations and restrictions. One of the significant parameters to optimize in any MapReduce job is communication cost between the map and reduce phases. The communication cost can be optimized by minimizing the number of copies of inputs sent to the reducers. The communication cost is closely related to the number of reducers of constrained capacity that are used to accommodate appropriately the inputs, so that the requirement of how the inputs must meet in a reducer is satisfied. In this work, we consider a family of problems where it is required that each input meets with each other input in at least one reducer. We also consider a slightly different family of problems in which, each input of a list, X, is required to meet each input of another list, Y, in at least one reducer. We prove that finding an optimal mapping schema for these families of problems is NP-hard, and present a bin-packing-based approximation algorithm for finding a near optimal mapping schema.Comment: This paper is accepted in ACM Transactions on Knowledge Discovery from Data (TKDD), August 2016. Preliminary versions of this paper have appeared in the proceeding of DISC 2014 and BeyondMR 201

On the Tomography of Networks and Multicast Trees

In this paper we model the tomography of scale free networks by studying the structure of layers around an arbitrary network node. We find, both analytically and empirically, that the distance distribution of all nodes from a specific network node consists of two regimes. The first is characterized by rapid growth, and the second decays exponentially. We also show that the nodes degree distribution at each layer is a power law with an exponential cut-off. We obtain similar results for the layers surrounding the root of multicast trees cut from such networks, as well as the Internet. All of our results were obtained both analytically and on empirical Interenet data

A Formal Approach to Exploiting Multi-Stage Attacks based on File-System Vulnerabilities of Web Applications (Extended Version)

Web applications require access to the file-system for many different tasks. When analyzing the security of a web application, secu- rity analysts should thus consider the impact that file-system operations have on the security of the whole application. Moreover, the analysis should take into consideration how file-system vulnerabilities might in- teract with other vulnerabilities leading an attacker to breach into the web application. In this paper, we first propose a classification of file- system vulnerabilities, and then, based on this classification, we present a formal approach that allows one to exploit file-system vulnerabilities. We give a formal representation of web applications, databases and file- systems, and show how to reason about file-system vulnerabilities. We also show how to combine file-system vulnerabilities and SQL-Injection vulnerabilities for the identification of complex, multi-stage attacks. We have developed an automatic tool that implements our approach and we show its efficiency by discussing several real-world case studies, which are witness to the fact that our tool can generate, and exploit, complex attacks that, to the best of our knowledge, no other state-of-the-art-tool for the security of web applications can find

On Byzantine Broadcast in Loosely Connected Networks

We consider the problem of reliably broadcasting information in a multihop asynchronous network that is subject to Byzantine failures. Most existing approaches give conditions for perfect reliable broadcast (all correct nodes deliver the authentic message and nothing else), but they require a highly connected network. An approach giving only probabilistic guarantees (correct nodes deliver the authentic message with high probability) was recently proposed for loosely connected networks, such as grids and tori. Yet, the proposed solution requires a specific initialization (that includes global knowledge) of each node, which may be difficult or impossible to guarantee in self-organizing networks - for instance, a wireless sensor network, especially if they are prone to Byzantine failures. In this paper, we propose a new protocol offering guarantees for loosely connected networks that does not require such global knowledge dependent initialization. In more details, we give a methodology to determine whether a set of nodes will always deliver the authentic message, in any execution. Then, we give conditions for perfect reliable broadcast in a torus network. Finally, we provide experimental evaluation for our solution, and determine the number of randomly distributed Byzantine failures than can be tolerated, for a given correct broadcast probability.Comment: 1

Superpatterns and Universal Point Sets

An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by forbidden patterns, and we construct superpatterns of size n^2/4 + Theta(n) for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct universal point sets of size n^2/4 - Theta(n), smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding permutations has superpatterns of size O(n log^O(1) n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets.Comment: GD 2013 special issue of JGA