Results 1  10
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49
Internet topology: connectivity of IP graphs
, 2001
"... 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 ASlevel connectivity. We describe data acquisition and construction of several IPlevel ..."
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Cited by 84 (6 self)
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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 ASlevel connectivity. We describe data acquisition and construction of several IPlevel 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.
Resolution lower bounds for perfect matching principles
 Journal of Computer and System Sciences
"... For an arbitrary hypergraph H, letPM(H) be the propositional formula asserting that H contains a perfect matching. We show that every resolution refutation of PM(H) musthavesize exp Ω δ(H) λ(H)r(H)(log n(H))(r(H)+logn(H)) where n(H) is the number of vertices, δ(H) is the minimal degree of a vertex, ..."
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Cited by 39 (4 self)
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For an arbitrary hypergraph H, letPM(H) be the propositional formula asserting that H contains a perfect matching. We show that every resolution refutation of PM(H) musthavesize exp Ω δ(H) λ(H)r(H)(log n(H))(r(H)+logn(H)) where n(H) is the number of vertices, δ(H) is the minimal degree of a vertex, r(H) is the maximal size of an edge, and λ(H) is the maximal number of edges incident to two different vertices. For ordinary graphs G our general bound considerably simplifies to exp Ω (implying an exp(Ω(δ(G) 1/3)) lower bound that depends on the minimal degree only). As a direct corollary, every resolution proof of the functional ( ( onto)) version of must have size exp Ω (which the pigeonhole principle onto − FPHP m n n (log m) 2 δ(G) (log n(G)) 2 becomes exp ( Ω(n 1/3) ) when the number of pigeons m is unbounded). This in turn immediately implies an exp(Ω(t/n 3)) lower bound on the size of resolution proofs of the principle asserting that the circuit size of the Boolean function fn in n variables is greater than t. Inparticular,Resolution does not possess efficient proofs of NP ⊆ P/poly. These results relativize, in a natural way, to a more general principle M(UH) asserting that H contains a matching covering all vertices in U ⊆ V (H).
Analysis of RouteViews BGP data: policy atoms
, 2001
"... 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 p ..."
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Cited by 31 (1 self)
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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.
Elliptic Curve Paillier Schemes
, 2001
"... . 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 ar ..."
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Cited by 16 (0 self)
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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.
Depth3 arithmetic formulae over fields of characteristic zero
 In CCC
, 1999
"... In this paper we prove near quadratic lower bounds for depth3 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 ..."
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Cited by 15 (2 self)
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In this paper we prove near quadratic lower bounds for depth3 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 depth3 arithmetic formulas and depth4 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 codimension. Lower bounds for related models utilize an algebraic analog of Nečhiporuk lower bound on Boolean formulae.
J.M.: All in the XL Family: Theory and Practice
 ICISC 2004. LNCS
, 2005
"... Abstract. The XL (eXtended Linearization) equationsolving algorithm belongs to the same extended family as the advanced Gröbner Bases methods F4/F5. XLanditsrelativesmaybeusedasdirectattacks against multivariate PublicKey Cryptosystems and as final stages for many “algebraic cryptanalysis ” used t ..."
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Cited by 13 (8 self)
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Abstract. The XL (eXtended Linearization) equationsolving algorithm belongs to the same extended family as the advanced Gröbner Bases methods F4/F5. XLanditsrelativesmaybeusedasdirectattacks against multivariate PublicKey Cryptosystems and as final stages for many “algebraic cryptanalysis ” used today. We analyze the applicability and performance of XL and its relatives, particularly for generic systems of equations over mediumsized finite fields. In examining the extended family of Gröbner Bases and XL from theoretical, empirical and practical viewpoints, we add to the general understanding of equationsolving. Moreover, we give rigorous conditions for the successful termination of XL, Gröbner Bases methods and relatives. Thus we have a better grasp of how such algebraic attacks should be applied. We also compute revised security estimates for multivariate cryptosystems. For example, the schemes SFLASH v2 and HFE Challenge 2 are shown to be unbroken by XL variants.
The stabilisation of quantum computations
 Proceedings of the Workshop on Physics and Computation, PhysComp '94 (Los Alamitos: IEEE Comp. Soc
, 1994
"... 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 netwo ..."
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Cited by 13 (1 self)
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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
On the existence of equiangular tight frames
 Linear Algebra Appl
, 2004
"... In a recent paper, Holmes and Paulsen established a necessary condition for the existence of an Nvector equiangular tight frame in a ddimensional real Euclidean space. This article develops much stronger necessary conditions using a combination of field theory and graph theory. This investigation ..."
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Cited by 12 (1 self)
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In a recent paper, Holmes and Paulsen established a necessary condition for the existence of an Nvector equiangular tight frame in a ddimensional real Euclidean space. This article develops much stronger necessary conditions using a combination of field theory and graph theory. This investigation rules out many possibilities admitted by the work of Holmes and Paulsen. Using a new onetoone correspondence between equivalence classes of real equiangular tight frames and strongly regular graphs of a certain type, it has been verified that a real equiangular tight frame exists for each pair (d, N) with N ≤ 100 that meets the new conditions. The arguments also extend to deliver novel necessary conditions for the existence of equiangular tight frames whose Gram matrices have entries drawn from a discrete set of complex numbers. 1