2,951 research outputs found

    Tight lower bounds on the ambiguity of strong, total, associative, one-way functions

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    AbstractWe study the ambiguity, or “many-to-one”-ness, of two-argument, one-way functions that are strong (that is, hard to invert even if one of their arguments is given), total, and associative. Such powerful one-way functions are the basis of a cryptographic paradigm described by Rabi and Sherman (Inform. Process. Lett. 64(2) (1997) 239) and were shown by Hemaspaandra and Rothe (J. Comput. System Sci. 58(3) (1999) 648) to exist exactly if standard one-way functions exist.Rabi and Sherman (1997) show that no total, associative function defined over a universe having at least two elements is one-to-one. We show that if P≠UP, then, for every d∈N+, there is an O(log1dn)-to-one, strong, total, associative, one-way function σd. We argue that this bound is tight in the sense that any total, associative function having similar properties to σd but not necessarily strong or one-way must have at least the same order of magnitude of ambiguity as σd has. We demonstrate that the techniques used in proving the above-stated results easily apply to other classes of total, associative functions.We provide a complete characterization for the existence of strong, total, associative, one-way functions whose ambiguity approaches the lower bounds we provide. We say a language is in PolylogP if there exists a polynomial-time Turing machine M accepting the language such that for some d∈R+ it holds that M has on each string x at most O(logdn) accepting paths, where n=|x|. We show that P≠PolylogP if and only for some d∈R+ there exists an O(logdn)-to-one, strong, total, associative, one-way function

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    Efficient enumeration of solutions produced by closure operations

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    In this paper we address the problem of generating all elements obtained by the saturation of an initial set by some operations. More precisely, we prove that we can generate the closure of a boolean relation (a set of boolean vectors) by polymorphisms with a polynomial delay. Therefore we can compute with polynomial delay the closure of a family of sets by any set of "set operations": union, intersection, symmetric difference, subsets, supersets 
\dots). To do so, we study the MembershipFMembership_{\mathcal{F}} problem: for a set of operations F\mathcal{F}, decide whether an element belongs to the closure by F\mathcal{F} of a family of elements. In the boolean case, we prove that MembershipFMembership_{\mathcal{F}} is in P for any set of boolean operations F\mathcal{F}. When the input vectors are over a domain larger than two elements, we prove that the generic enumeration method fails, since MembershipFMembership_{\mathcal{F}} is NP-hard for some F\mathcal{F}. We also study the problem of generating minimal or maximal elements of closures and prove that some of them are related to well known enumeration problems such as the enumeration of the circuits of a matroid or the enumeration of maximal independent sets of a hypergraph. This article improves on previous works of the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of the same name which appeared in STACS 2016. Final version for DMTCS journa

    Cell-Probe Lower Bounds from Online Communication Complexity

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    In this work, we introduce an online model for communication complexity. Analogous to how online algorithms receive their input piece-by-piece, our model presents one of the players, Bob, his input piece-by-piece, and has the players Alice and Bob cooperate to compute a result each time before the next piece is revealed to Bob. This model has a closer and more natural correspondence to dynamic data structures than classic communication models do, and hence presents a new perspective on data structures. We first present a tight lower bound for the online set intersection problem in the online communication model, demonstrating a general approach for proving online communication lower bounds. The online communication model prevents a batching trick that classic communication complexity allows, and yields a stronger lower bound. We then apply the online communication model to prove data structure lower bounds for two dynamic data structure problems: the Group Range problem and the Dynamic Connectivity problem for forests. Both of the problems admit a worst case O(log⁡n)O(\log n)-time data structure. Using online communication complexity, we prove a tight cell-probe lower bound for each: spending o(log⁡n)o(\log n) (even amortized) time per operation results in at best an exp⁡(−ή2n)\exp(-\delta^2 n) probability of correctly answering a (1/2+ή)(1/2+\delta)-fraction of the nn queries

    Random Neural Networks and Optimisation

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    In this thesis we introduce new models and learning algorithms for the Random Neural Network (RNN), and we develop RNN-based and other approaches for the solution of emergency management optimisation problems. With respect to RNN developments, two novel supervised learning algorithms are proposed. The first, is a gradient descent algorithm for an RNN extension model that we have introduced, the RNN with synchronised interactions (RNNSI), which was inspired from the synchronised firing activity observed in brain neural circuits. The second algorithm is based on modelling the signal-flow equations in RNN as a nonnegative least squares (NNLS) problem. NNLS is solved using a limited-memory quasi-Newton algorithm specifically designed for the RNN case. Regarding the investigation of emergency management optimisation problems, we examine combinatorial assignment problems that require fast, distributed and close to optimal solution, under information uncertainty. We consider three different problems with the above characteristics associated with the assignment of emergency units to incidents with injured civilians (AEUI), the assignment of assets to tasks under execution uncertainty (ATAU), and the deployment of a robotic network to establish communication with trapped civilians (DRNCTC). AEUI is solved by training an RNN tool with instances of the optimisation problem and then using the trained RNN for decision making; training is achieved using the developed learning algorithms. For the solution of ATAU problem, we introduce two different approaches. The first is based on mapping parameters of the optimisation problem to RNN parameters, and the second on solving a sequence of minimum cost flow problems on appropriately constructed networks with estimated arc costs. For the exact solution of DRNCTC problem, we develop a mixed-integer linear programming formulation, which is based on network flows. Finally, we design and implement distributed heuristic algorithms for the deployment of robots when the civilian locations are known or uncertain
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