4,210 research outputs found

    Stopping time signatures for some algorithms in cryptography

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    We consider the normalized distribution of the overall running times of some cryptographic algorithms, and what information they reveal about the algorithms. Recent work of Deift, Menon, Olver, Pfrang, and Trogdon has shown that certain numerical algorithms applied to large random matrices exhibit a characteristic distribution of running times, which depends only on the algorithm but are independent of the choice of probability distributions for the matrices. Different algorithms often exhibit different running time distributions, and so the histograms for these running time distributions provide a time-signature for the algorithms, making it possible, in many cases, to distinguish one algorithm from another. In this paper we extend this analysis to cryptographic algorithms, and present examples of such algorithms with time-signatures that are indistinguishable, and others with time-signatures that are clearly distinct.Comment: 20 page

    Testing Small Set Expansion in General Graphs

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    We consider the problem of testing small set expansion for general graphs. A graph GG is a (k,ϕ)(k,\phi)-expander if every subset of volume at most kk has conductance at least ϕ\phi. Small set expansion has recently received significant attention due to its close connection to the unique games conjecture, the local graph partitioning algorithms and locally testable codes. We give testers with two-sided error and one-sided error in the adjacency list model that allows degree and neighbor queries to the oracle of the input graph. The testers take as input an nn-vertex graph GG, a volume bound kk, an expansion bound ϕ\phi and a distance parameter ε>0\varepsilon>0. For the two-sided error tester, with probability at least 2/32/3, it accepts the graph if it is a (k,ϕ)(k,\phi)-expander and rejects the graph if it is ε\varepsilon-far from any (k,ϕ)(k^*,\phi^*)-expander, where k=Θ(kε)k^*=\Theta(k\varepsilon) and ϕ=Θ(ϕ4min{log(4m/k),logn}(lnk))\phi^*=\Theta(\frac{\phi^4}{\min\{\log(4m/k),\log n\}\cdot(\ln k)}). The query complexity and running time of the tester are O~(mϕ4ε2)\widetilde{O}(\sqrt{m}\phi^{-4}\varepsilon^{-2}), where mm is the number of edges of the graph. For the one-sided error tester, it accepts every (k,ϕ)(k,\phi)-expander, and with probability at least 2/32/3, rejects every graph that is ε\varepsilon-far from (k,ϕ)(k^*,\phi^*)-expander, where k=O(k1ξ)k^*=O(k^{1-\xi}) and ϕ=O(ξϕ2)\phi^*=O(\xi\phi^2) for any 0<ξ<10<\xi<1. The query complexity and running time of this tester are O~(nε3+kεϕ4)\widetilde{O}(\sqrt{\frac{n}{\varepsilon^3}}+\frac{k}{\varepsilon \phi^4}). We also give a two-sided error tester with smaller gap between ϕ\phi^* and ϕ\phi in the rotation map model that allows (neighbor, index) queries and degree queries.Comment: 23 pages; STACS 201

    On sampling nodes in a network

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    Random walk is an important tool in many graph mining applications including estimating graph parameters, sampling portions of the graph, and extracting dense communities. In this paper we consider the problem of sampling nodes from a large graph according to a prescribed distribution by using random walk as the basic primitive. Our goal is to obtain algorithms that make a small number of queries to the graph but output a node that is sampled according to the prescribed distribution. Focusing on the uniform distribution case, we study the query complexity of three algorithms and show a near-tight bound expressed in terms of the parameters of the graph such as average degree and the mixing time. Both theoretically and empirically, we show that some algorithms are preferable in practice than the others. We also extend our study to the problem of sampling nodes according to some polynomial function of their degrees; this has implications for designing efficient algorithms for applications such as triangle counting

    Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions

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    We introduce spatially explicit stochastic processes to model multispecies host-symbiont interactions. The host environment is static, modeled by the infinite percolation cluster of site percolation. Symbionts evolve on the infinite cluster through contact or voter type interactions, where each host may be infected by a colony of symbionts. In the presence of a single symbiont species, the condition for invasion as a function of the density of the habitat of hosts and the maximal size of the colonies is investigated in details. In the presence of multiple symbiont species, it is proved that the community of symbionts clusters in two dimensions whereas symbiont species may coexist in higher dimensions.Comment: Published in at http://dx.doi.org/10.1214/10-AAP734 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org
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