Solving Low Density Knapsacks

Brickell, Ernest F.

doi:10.1007/978-1-4684-4730-9_2

Cited by 64 publications

(79 citation statements)

References 5 publications

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“…The analysis of [13] showed that availability of such an oracle would let the Lagarias-Odlyzko algorithm solve almost all subset sum problems of density < 0.6463..., but not higher than that. (Similar analyses are not available for the Brickell algorithm [1], although it seems to require even lower densities. See also [9].)…”

Section: Introductionmentioning

confidence: 99%

Improved low-density subset sum algorithms

et al. 1992

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Abstract. The general subset sum problem is NP-complete. However, there are two algorithms, one due to Brickell and the other to Lagarias and Odlyzko, which in polynomial time solve almost all subset sum problems of sufficiently low density. Both methods rely on basis reduction algorithms to find short non-zero vectors in special lattices. The Lagarias-Odlyzko algorithm would solve almost all subset sum problems of density < 0.6463... in polynomial time if it could invoke a polynomial-time algorithm for finding the shortest non-zero vector in a lattice. This paper presents two modifications of that algorithm, either one of which would solve almost all problems of density < 0.9408... if it could find shortest non-zero vectors in lattices. These modifications also yield dramatic improvements in practice when they are combined with known lattice basis reduction algorithms.

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Section: Introductionmentioning

confidence: 99%

Improved low-density subset sum algorithms

et al. 1992

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“…There can be two cases: a) D≤1: These low-density subset-sum problems are efficiently solved by reduction to a short vector in a lattice, as presented by Brickell [4]; Lagarias and Odlyzko [5]; Martello and Toth [6]; Coster et al [7]. b) D>1: These medium and high-density subset-sum problems are solvable by dynamic programming techniques or using analytical number theory, such as those presented in Chaimovich et al [8]; Galil and Margalit [9]; Flaxman and Przydatek [10]; with some of these failing to find a solution if certain bounds for n or R are not respected.…”

Section: Related Workmentioning

confidence: 99%

A Fast Heuristic Algorithm for Solving High-Density Subset-Sum Problems

Nag¹

2017

IJMSC

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The subset sum problem is to decide whether for a given set of integers A and an integer S, a possible subset of A exists such that the sum of its elements is equal to S. The problem of determining whether such a subset exists is NP-complete; which is the basis for cryptosystems of knapsack type. In this paper a fast heuristic algorithm is proposed for solving subset sum problems in pseudo-polynomial time. Extensive computational evidence suggests that the algorithm almost always finds a solution to the problem when one exists. The runtime performance of the algorithm is also analyzed.

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“…These have all been analyzed and broken, generally using the same cryptographic techniques, and litter the cryptographic highway [260,253,269,921,15,919,920,922,366,254,263,255]. Good overviews of these systems and their cryptanalyses can be found in [267,479,257,268].…”

Section: Knapsack Variantsmentioning

confidence: 99%

Applied cryptography: Protocols, algorithms, and source code in C

1994

Computer Law & Security Review

771

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Solving Low Density Knapsacks

Cited by 64 publications

References 5 publications

Improved low-density subset sum algorithms

Improved low-density subset sum algorithms

A Fast Heuristic Algorithm for Solving High-Density Subset-Sum Problems

Applied cryptography: Protocols, algorithms, and source code in C

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