AnO(n⋅2) time and O(k⋅2) space algorithm for certain np-complete problems

Vyskoc, Jozef

doi:10.1016/0304-3975(90)90127-4

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2012

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“…The application of these algorithms to the fixed weight setting relies on splitting systems to perform the necessary decomposition. Another candidate for the k-division algorithm is the generalization of Schroeppel-Shamir outlined in [19] (but see [3] for a rebuttal).…”

Section: Prior Work and New Resultsmentioning

confidence: 99%

Division algorithms for the fixed weight subset sum problem

Shallue

2012

Preprint

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Given positive integers a 1 , . . . , an, t, the fixed weight subset sum problem is to find a subset of the a i that sum to t, where the subset has a prescribed number of elements. It is this problem that underlies the security of modern knapsack cryptosystems, and solving the problem results directly in a message attack. We present new exponential algorithms that do not rely on lattices, and hence will be applicable when lattice basis reduction algorithms fail. These algorithms rely on a generalization of the notion of splitting system given by Stinson [18]. In particular, if the problem has length n and weight ℓ then for constant k a power of two less than n we apply a k-set birthday algorithm to the splitting system of the problem. This randomized algorithm has time and space complexity that satisfies T • S log k = O( n ℓ ) (where the constant depends uniformly on k). In addition to using space efficiently, the algorithm is highly parallelizable.

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Section: Prior Work and New Resultsmentioning

confidence: 99%