An Improved Low-Density Subset Sum Algorithm

Coster, M.J.; LaMacchia, Brian A.; Odlyzko, Andrew; Schnorr, Claus-Peter

doi:10.1007/3-540-46416-6_4

Cited by 59 publications

(65 citation statements)

References 16 publications

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“…Their attack does not incorporate the weakness on the trapdoor itself, rather than only using the fact that the knapsack problems produced are generally weaker that a random one. This result was subsequently improved in [25,24,66,58]. Nevertheless, some improvements of knapsack cryptosystems were also proposed (e.g.…”

Section: Knapsack Cryptosystemsmentioning

confidence: 99%

Recursive Lattice Reduction

Plantard

Susilo

2010

Lecture Notes in Computer Science

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Abstract. Lattice reduction is known to be a very powerful tool in modern cryptanalysis. In the literature, there are many lattice reduction algorithms that have been proposed with various time complexity (from quadratic to subexponential). These algorithms can be utilized to find a short vector of a lattice with a small norm. Over time, shorter vector will be found by incorporating these methods. In this paper, we take a different approach by presenting a methodology that can be applied to any lattice reduction algorithms, with the implication that enables us to find a shorter vector (i.e. a smaller solution) while requiring shorter computation time. Instead of applying a lattice reduction algorithm to a complete lattice, we work on a sublattice with a smaller dimension chosen in the function of the lattice reduction algorithm that is being used. This way, the lattice reduction algorithm will be fully utilized and hence, it will produce a better solution. Furthermore, as the dimension of the lattice becomes smaller, the time complexity will be better. Hence, our methodology provides us with a new direction to build a lattice that is resistant to lattice reduction attacks. Moreover, based on this methodology, we also propose a recursive method for producing an optimal approach for lattice reduction with optimal computational time, regardless of the lattice reduction algorithm used. We evaluate our technique by applying it to break the lattice challenge 1 by producing the shortest vector known so far. Our results outperform the existing known results and hence, our results achieve the record in the lattice challenge problem.

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Section: Knapsack Cryptosystemsmentioning

confidence: 99%

Recursive Lattice Reduction

Plantard

Susilo

2010

Lecture Notes in Computer Science

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“…We have chosen knapsack lattices [19,9,8,28,10] and random lattices (as defined by Goldstein and Mayer [12,27]) as examples for sparse lattice bases. Unimodular lattice bases serve as representatives for dense lattice bases Previous experiments [37,2] showed that unimodular lattice bases are more difficult to reduce than, e.g., knapsack lattices given the same dimension and maximum length of the basis entries.…”

Section: Lattice Basesmentioning

confidence: 99%

Parallel Lattice Basis Reduction Using a Multi-threaded Schnorr-Euchner LLL Algorithm

Backes

Wetzel

2009

Lecture Notes in Computer Science

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Abstract. In this paper, we introduce a new parallel variant of the LLL lattice basis reduction algorithm. Our new, multi-threaded algorithm is the first to provide an efficient, parallel implementation of the Schorr-Euchner algorithm for today's multi-processor, multi-core computer architectures. Experiments with sparse and dense lattice bases show a speed-up factor of about 1.8 for the 2-thread and about factor 3.2 for the 4-thread version of our new parallel lattice basis reduction algorithm in comparison to the traditional non-parallel algorithm.

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“…when the function is used in expansion mode). According to [9], the best known provable lattice attack of this type [4] succeeds with high probability over a random choice of a[1], . .…”

Section: Definition 4 (Subset Sum Problemmentioning

confidence: 99%

“…and try to use the method of [4] to find a solution involving only those integers (i.e. set the n − n remaining weights to zero).…”

Section: Definition 4 (Subset Sum Problemmentioning

confidence: 99%

Higher Order Universal One-Way Hash Functions from the Subset Sum Assumption

Steinfeld

Pieprzyk

Wang

2006

Public Key Cryptography - PKC 2006

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Abstract. Universal One-Way Hash Functions (UOWHFs) may be used in place of collision-resistant functions in many public-key cryptographic applications. At Asiacrypt 2004, Hong, Preneel and Lee introduced the stronger security notion of higher order UOWHFs to allow construction of long-input UOWHFs using the Merkle-Damgård domain extender. However, they did not provide any provably secure constructions for higher order UOWHFs.We show that the subset sum hash function is a kth order Universal One-Way Hash Function (hashing n bits to m < n bits) under the Subset Sum assumption for k = O(log m). Therefore we strengthen a previous result of Impagliazzo and Naor, who showed that the subset sum hash function is a UOWHF under the Subset Sum assumption. We believe our result is of theoretical interest; as far as we are aware, it is the first example of a natural and computationally efficient UOWHF which is also a provably secure higher order UOWHF under the same well-known cryptographic assumption, whereas this assumption does not seem sufficient to prove its collision-resistance. A consequence of our result is that one can apply the Merkle-Damgård extender to the subset sum compression function with 'extension factor' k + 1, while losing (at most) about k bits of UOWHF security relative to the UOWHF security of the compression function. The method also leads to a saving of up to m log(k + 1) bits in key length relative to the Shoup XOR-Mask domain extender applied to the subset sum compression function.

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An Improved Low-Density Subset Sum Algorithm

Cited by 59 publications

References 16 publications

Recursive Lattice Reduction

Recursive Lattice Reduction

Parallel Lattice Basis Reduction Using a Multi-threaded Schnorr-Euchner LLL Algorithm

Higher Order Universal One-Way Hash Functions from the Subset Sum Assumption

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