Constructions of Sparse Asymmetric Connectors

Baltz, Andreas; Jäger, Gerold; Srivastav, Anand

doi:10.1007/978-3-540-24597-1_2

Cited by 4 publications

(9 citation statements)

References 7 publications

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“…Hwang and Richards [7] gave an explicit construction for (n, N, 2)-connectors of size (1+o(1))N √ n if n ≤ √ N . Baltz, Jäger and Srivastav [1], [2] showed by a probabilistic argument the existence of (n, N, 2)-connectors of size O(N ), if n ≤ N 1/2−ε , ε > 0. They also extended the result of Hwang and Richards and improved it by a constant factor.…”

Section: Introductionmentioning

confidence: 99%

“…Such connectors with d = 2 are of particular interest (refering to [1,2]) in the design of sparse electronic switches. A challenging problem is to construct linear-sized (n, N, 2)-connectors.…”

Section: Introductionmentioning

confidence: 99%

“…We consider the problem of construction of sparse (n, N, 2)-connectors (depth 2 connectors) when n N . The probabilistic argument in [1] shows the existence of (n, N, 2)-connectors of size (number of edges) O(N ) if n ≤ N 1/2−ε , ε > 0. However, the known explicit constructions with…”

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confidence: 99%

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Sparse Asymmetric Connectors in Communication Networks

Ahlswede

Aydinian

2006

Lecture Notes in Computer Science

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Abstract. An (n, N, d)-connector is an acyclic digraph with n inputs and N outputs in which for any injective mapping of input vertices into output vertices there exist n vertex disjoint paths of length d joining each input to its corresponding output. We consider the problem of construction of sparse (n, N, 2)-connectors (depth 2 connectors) when n N . The probabilistic argument in [1] shows the existence of (n, N, 2)-connectors of size (number of edges) O(N ) if n ≤ N 1/2−ε , ε > 0. However, the known explicit constructions withHere we present a simple combinatorial construction for (n, N, 2)-connectors of size O(N log 2 n). We also consider depth 2 faulttolerant connectors under arc or node failures.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sparse Asymmetric Connectors in Communication Networks

Ahlswede

Aydinian

2006

Lecture Notes in Computer Science

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“…Beyond the theoretical interest in the question, the need for such systems occurs in reality, e.g., in multicast switch design [K03]. This motivated the work of Baltz, Jäger, and Srivastav [BJS03] who gave explicit (N, K) depth two connectors with O(N √ K) edges, for K ≤ √ N . This was followed by an entirely different combinatorial approach of Ahlswede and Aydinian [AA03], based on the Kruskal-Katona theorem, that yields explicit depth two connectors with O(N log K) edges, given that K ≤ N O(1/ √ log N ) .…”

Section: Introductionmentioning

confidence: 99%

Constructions of sparse asymmetric connectors with number theoretic methods

Baltz¹,

Jäger²,

Srivastav³

2005

Networks

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An (N, K, ) connector (rearrangeable network) is a digraph with N inputs and K outputs such that for any injective mapping from the outputs to the inputs there is a set of vertex disjoint paths of length implementing it. The study of this problem begins with the pioneering work of Shannon, Slepian, Clos and Beneš in the 1950's who studied the balanced N = K case, and arbitrary depth , and later on by Pippenger and Yao who used expanders to construct small depth, balanced connectors.In the last decade there has been research on the unbalanced case. This started with Oruç who showed an explicit construction with O(N ) edges, for K ≤ N log(N ) , and proved some lower bounds on small depth networks. Later on Baltz, Jäger, and Srivastav suggested an explicit depth two construction for the K ≤ √ N case with O(N √ K) edges, which was followed by an entirely different depth two construction by Ahlswede and Aydinian with O(N log K) edges, but which worked only for K ≤ N O(1/ √ log N ) . We show that non-constructively depth two connectors with O(N ) edges exist for K ≤ N 1/2− , for every constant > 0. Furthermore, for that range we show an explicit construction with only N poly(log N ) edges.The solution is based on new expander techniques that were developed in the study of extractors. We believe that these new tools can be very useful in other network constructions as well. We feel that despite the success of these newer tools in theoretical computer science they are still widely neglected in network design, partly because they are so new. We hope our work will find other applications in network design.

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“…Such a proof implies the absence of a generic (black-box) attack against C(F), i.e., an attack which does not exploit specific properties of F, but uses it merely as a black-box. 3 Such a generic proof is not an ultimate security proof for C(F), but it proves that the construction C(·) itself has no weakness. A main advantage of such a proof is that it applies to every cryptographic property of interest (which a random function has), not just to specific properties like collision-resistance.…”

Section: Significance Of Domain Extension For Public Random Functionsmentioning

confidence: 99%

Domain Extension of Public Random Functions: Beyond the Birthday Barrier

Maurer

Tessaro

Advances in Cryptology - CRYPTO 2007

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A public random function is a random function that is accessible by all parties, including the adversary. For example, a (public) random oracle is a public random function {0, 1} * → {0, 1} n . The natural problem of constructing a public random oracle from a public random function {0, 1} m → {0, 1} n (for some m > n) was first considered at Crypto 2005 by Coron et al. who proved the security of variants of the Merkle-Damgård construction against adversaries issuing up to O(2 n/2 ) queries to the construction and to the underlying compression function. This bound is less than the square root of n2 m , the number of random bits contained in the underlying random function.In this paper, we investigate domain extenders for public random functions approaching optimal security. In particular, for all ∈ (0, 1) and all functions m and (polynomial in n), we provide a construction C ,m, (·) which extends a public random function R : {0, 1} n → {0, 1} n to a function C ,m, (R) : {0, 1} m(n) → {0, 1} (n) with time-complexity polynomial in n and 1/ and which is secure against adversaries which make up to Θ(2 n(1− ) ) queries. A central tool for achieving high security are special classes of unbalanced bipartite expander graphs with small degree. The achievability of practical (as opposed to complexity-theoretic) efficiency is proved by a non-constructive existence proof.Combined with the iterated constructions of Coron et al., our result leads to the first iterated construction of a hash function {0, 1} * → {0, 1} n from a component function {0, 1} n → {0, 1} n that withstands all recently proposed generic attacks against iterated hash functions, like Joux's multi-collision attack, Kelsey and Schneier's second-preimage attack, and Kelsey and Kohno's herding attacks.

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Constructions of Sparse Asymmetric Connectors

Cited by 4 publications

References 7 publications

Sparse Asymmetric Connectors in Communication Networks

Sparse Asymmetric Connectors in Communication Networks

Constructions of sparse asymmetric connectors with number theoretic methods

Domain Extension of Public Random Functions: Beyond the Birthday Barrier

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