We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with link-costs and, for each pair fi; jg of nodes, an edge-connectivity requirement r ij . The goal is to find a minimum-cost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within 2dlog 2 (r + 1)e of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial min-max approximate equality relating minimum-cost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.

We present a randomized 2 O(n) time algorithm to compute a shortest non-zero vector in an n-dimensional rational lattice. The best known time upper bound for this problem was 2 O(n log n)

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A quorum system is a collection of sets (quorums) every two of which intersect. Quorum systems have been used for many applications in the area of distributed systems, including mutual exclusion, data replication and dissemination of information Given a strategy to pick quorums, the load L(S) is the minimal access probability of the busiest element, minimizing over the strategies. The capacity Cap(S) is the highest quorum accesses rate that S can handle, so Cap(S) = 1=L(S).

We show that if the private exponent d used in the RSA public-key cryptosystem is less than N^0.292 then the system is insecure. This is the first improvement over an old result of Wiener showing that when d is less than N^0.25 the RSA system is insecure. We hope our approach can be used to eventually improve the bound to d less than N^0.5.

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We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higher-dimensional Dedekind sums, complexity of the Presburger arithmetic, efficient computations with rational functions, and others. Although the main slant is algorithmic, structural results are discussed, such as relations to the general theory of valuations on polyhedra and connections with the theory of toric varieties. The paper surveys known results and presents some new results and connections.

Lattices are regular arrangements of points in n-dimensional space, whose study appeared in the 19th century in both number theory and crystallography. Since the appearance of the celebrated LenstraLenstra -Lov'asz lattice basis reduction algorithm twenty years ago, lattices have had surprising applications in cryptology. Until recently, the applications of lattices to cryptology were only negative, as lattices were used to break various cryptographic schemes. Paradoxically, several positive cryptographic applications of lattices have emerged in the past five years: there now exist public-key cryptosystems based on the hardness of lattice problems, and lattices play a crucial role in a few security proofs.

We improve a connection of the worst-case complexity and the average-case complexity of some well-known lattice problems. This fascinating connection was first discovered by Ajtai [1] in 1996. We improve the exponent of this connection from 8 to 3:5 + ffl. Department of Computer Science, State University of New York at Buffalo, Buffalo, NY 14260. Research supported in part by NSF grants CCR-9319393 and CCR-9634665, and an Alfred P. Sloan Fellowship. Email: cai@cs.buffalo.edu y Department of Computer Science, State University of New York at Buffalo, Buffalo, NY 14260. Research supported in part by NSF grants CCR-9319393 and CCR-9634665. Email: apn@cs.buffalo.edu 1 Introduction A lattice L is a discrete additive subgroup of R n . There are many fascinating problems concerning lattices, both from a structural and from an algorithmic point of view [12, 20, 11, 13]. The study of lattice problems can be traced back to Gauss, Dirichlet and Hermite, among others [8, 6, 14]. The subje...

this article - Algorithmic Geometry of Numbers. The fundamental basis reduction algorithm of Lov'asz which first appeared in Lenstra, Lenstra, Lov'asz [46] was used in Lenstra's algorithm for integer programming and has since been applied in myriad contexts-starting with factorization of polynomials (A.K. Lenstra, [45]). Classical Geometry of Numbers has a special feature in that it studies the geometric properties of (convex) sets like volume, width etc. which come from the realm of continuous mathematics in relation to lattices which are discrete objects. This makes it ideal for applications to integer programming and other discrete optimization problems which seem inherently harder than their "continuous" counterparts like linear programming. 1