In this paper we introduce a framework for analyzing local properties of Internet connectivity. We compare BGP and probed topology data, finding that currently probed topology data yields much denser coverage of AS-level connectivity. We describe data acquisition and construction of several IP-level graphs derived from a collection of 220M skitter traceroutes. We find that a graph consisting of IP nodes and links contains 90.5% of its 629K nodes in the acyclic subgraph. In particular, 55% of the IP nodes are in trees. Full bidirectional connectivity is observed for a giant component containing 8.3% of IP nodes.

For an arbitrary hypergraph H, let PM(H) be the propositional formula asserting that contains a perfect matching. We show that every resolution refutation of PM(H) must have size #(H)r(H)(log n(H))(r(H) + log n(H)) where n(H) is the number of vertices, #(H) is the minimal degree of a vertex, r(H) is the maximal size of an edge, and #(H) is the maximal number of edges incident to two di#erent vertices.

In this paper we introduce a framework for analyzing BGP connectivity, and evaluate a number of new complexity measures for a union of core backbone BGP tables. Sensitive to resource limitations of router memory and CPU cycles, we focus on techniques to estimate redundancy of the merged tables, in particular how many entries are essential for complete and correct routing.

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. This paper is concerned with generalisations of Paillier's probabilistic encryption scheme from the integers modulo a square to elliptic curves over rings. Paillier himself described two public key encryption schemes based on anomalous elliptic curves over rings. It is argued that these schemes are not secure. A more natural generalisation of Paillier's scheme to elliptic curves is given.

We propose a method for the stabilisation of quantum computations (including quantum state storage). The method is based on the operation of projection into SYM, the symmetric subspace of the full state space of R redundant copies of the computer. We describe an efficient algorithm and quantum network effecting SYM–projection and discuss the stabilising effect of the proposed method in the context of unitary errors generated by hardware imprecision, and nonunitary errors arising from external environmental interaction. Finally, limitations of the method are discussed. Any realistic model of computation must conform to certain requirements imposed not by the mathematical properties of the model but by the laws of physics. Computations which require an exponentially increasing precision or exponential amount of time, space, energy or any other physical resource are normally regarded as unrealistic

. Finding a natural meeting ground between the highly developed complexity theory of computer science ---with its historical roots in logic and the discrete mathematics of the integers--- and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ---with its rich history and longstanding traditions in the continuous mathematics of analysis--- presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the Letts-Villard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...

In this paper we prove near quadratic lower bounds for depth-3 arithmetic formulae over fields of characteristic zero. Such bounds are obtained for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant. As corollaries we get the first nontrivial lower bounds for computing polynomials of constant degree, and a gap between the power depth-3 arithmetic formulas and depth-4 arithmetic formulas. The main technical contribution relates the complexity of computing a polynomial in this model to the wealth of partial derivatives it has on every affine subspace of small co-dimension. Lower bounds for related models utilize an algebraic analog of Nečhiporuk lower bound on Boolean formulae.

We propose a method for the stabilisation of quantum computations (including quantum state storage). The method is based on the operation of projection into SYM, the symmetric subspace of the full state space of R redundant copies of the computer. We describe an efficient algorithm and quantum network effecting SYM--projection and discuss the stabilising effect of the proposed method in the context of unitary errors generated by hardware imprecision, and nonunitary errors arising from external environmental interaction. Finally, limitations of the method are discussed. x1 Introduction Any realistic model of computation must conform to certain requirements imposed not by the mathematical properties of the model but by the laws of physics. Computations which require an exponentially increasing precision or exponential amount of time, space, energy or any other physical resource are normally regarded as unrealistic and intractable. Any actual computational process is subject to unavoid...

. Given an expression E using +; \Gamma; ; =; with operands from Z and from the set of real roots of integers, we describe a probabilistic algorithm that decides whether E = 0. The algorithms has a one-sided error. If E = 0, then the algorithm will give the correct answer. If E 6= 0, then the error probability can be made arbitrarily small. The algorithm has been implemented and is expected to be practical. Author's address: Johannes Bl¨omer Institute for Theoretical Computer Science ETH Zentrum CH-8092 Zurich Switzerland Technical Report #299, Departement Informatik, ETH Z¨urich. Electronically available from: ftp://ftp.inf.ethz.ch/pub/publications/tech-reports/ 1 Introduction In this paper we consider the following problem. Given a real radical expression without nested roots, that is, an expression E defined with operators +; \Gamma; ; =, with integer operands and operands of the form d p n; d 2 N;n 2 Z; d p n 2 R. We want to decide whether the expression E is zero. We desc...