Solving Medium-Density Subset Sum Problems in Expected Polynomial Time

Flaxman, Abraham D.; Przydatek, Bartosz

doi:10.1007/978-3-540-31856-9_25

Cited by 24 publications

(25 citation statements)

References 9 publications

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“…. , p δ } n of n-dimensional vectors with entries much smaller than p. Unfortunately, as for the case of the integer compact knapsack problem described above, this straightforward construction admits much faster solutions than exhaustive search: the resulting cc 16 (2007) Generalized compact knapsacks 369 generalized compact knapsack is equivalent to n independent instances of the knapsack problem modulo p, which can be efficiently solved in the worst case for any polynomially bounded p(n) = n O(1) by dynamic programming, and on the average for p(n) = n O(log n) using the methods of Flaxman & Przydatek (2005) and Lyubashevsky (2005).…”

Section: Generalized Compact Knapsacksmentioning

confidence: 99%

Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions

2007

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Abstract.We investigate the average-case complexity of a generalization of the compact knapsack problem to arbitrary rings: given m (random) ring elements a 1 , . . . , a m ∈ R and a (random) target value b ∈ R, find coefficients x 1 , . . . , x m ∈ S (where S is an appropriately chosen subset of R) such thatWe consider compact versions of the generalized knapsack where the set S is large and the number of weights m is small. Most variants of this problem considered in the past (e.g., when R = Z is the ring of the integers) can be easily solved in polynomial time even in the worst case. We propose a new choice of the ring R and subset S that yields generalized compact knapsacks that are seemingly very hard to solve on the average, even for very small values of m. Namely, we prove that for any unbounded function m = ω(1) with arbitrarily slow growth rate, solving our generalized compact knapsack problems on the average is at least as hard as the worst-case instance of various approximation problems over cyclic lattices. Specific worst-case lattice problems considered in this paper are the shortest independent vector problem SIVP and the guaranteed distance decoding problem GDD (a variant of the closest vector problem, CVP) for approximation factors n 1+ almost linear in the dimension of the lattice. Our results yield very efficient and provably secure one-way functions (based on worst-case complexity assumptions) with key size and time complexity almost linear in the security parameter n. Previous constructions with similar security guarantees required quadratic key size and computation time. Our results can also be formulated as a connection between the worst-case and average-case complexity of various lattice problems over cyclic and quasi-cyclic lattices.

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Section: Generalized Compact Knapsacksmentioning

confidence: 99%

Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions

2007

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“…The hardness of breaking SS(n, M ) depends on the ratio between n and log M , which is usually referred to as the density of the subset sum instance. When n/log M is less than 1/n or larger than n/log 2 n, the problem can be solved in polynomial time [LO85,Fri86,FP05,Lyu05,Sha08]. However, when the density is constant or even as small as O(1/log n), there are currently no algorithms that require less than 2 Ω(n) time.…”

Section: Introductionmentioning

confidence: 99%

Public-Key Cryptographic Primitives Provably as Secure as Subset Sum

Lyubashevsky

Palacio

Segev

2010

Lecture Notes in Computer Science

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Abstract. We propose a semantically-secure public-key encryption scheme whose security is polynomial-time equivalent to the hardness of solving random instances of the subset sum problem. The subset sum assumption required for the security of our scheme is weaker than that of existing subset-sum based encryption schemes, namely the lattice-based schemes of Ajtai and Dwork (STOC'97), Regev (STOC'03, STOC'05), and Peikert (STOC'09). Additionally, our proof of security is simple and direct. We also present a natural variant of our scheme that is secure against key-leakage attacks, and an oblivious transfer protocol that is secure against semi-honest adversaries.

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“…More sophisticated methods can improve the running time, for example [1] achieved a running time of O(n 7/4 / log 3/4 n) for instances with m log m = Θ(n 2 ). In [4], the range of problems solvable in polynomial time was extended to cases with m = 2 O(log n) 2 …”

Section: Previous Work and Resultsmentioning

confidence: 99%

An Improved Multi-set Algorithm for the Dense Subset Sum Problem

Shallue

2008

Lecture Notes in Computer Science

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Solving Medium-Density Subset Sum Problems in Expected Polynomial Time

Cited by 24 publications

References 9 publications

Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions

Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions

Public-Key Cryptographic Primitives Provably as Secure as Subset Sum

An Improved Multi-set Algorithm for the Dense Subset Sum Problem

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