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## Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems. (1993)

Venue: | Math. Programming |

Citations: | 325 - 6 self |

### Citations

14083 | Computers and Intractability: A Guide to the Theory of NP-completeness - Garey, Johnson - 1979 |

318 | Integer programming with a fixed number of variables. - Lenstra - 1983 |

124 | Solving low-density subset sum problems,” in
- Lagarias, Odlyzko
- 1983
(Show Context)
Citation Context ...apsack or subset sum problem is to solve, given positive integers a 1 ; : : : ; a n and s, the equation n X i=1 a i x i = s with x 1 ; : : : ; x n 2 f0; 1g: The Brickell [1] and the Lagarias--Odlyzko =-=[14]-=- algorithms solve almost all subset sum problems for which the density d = n = log 2 maxa i is sufficiently small. Radziszowski and Kreher [17] evaluate the performance of an improved variant of the L... |

106 | Improved low-density subset sum algorithms.
- Coster, Joux, et al.
- 1992
(Show Context)
Citation Context ...n proved rigorously that for almost all subset sum problems with density less that 0:9408 the shortest non--zero vector in the associated lattice basis (1) yields a solution of the subset sum problem =-=[3]-=-. In section 6 we describe a particular practical algorithm for block Korkin-- Zolotarev reduction. Using the improved reduction algorithms we can solve a much larger class of subset sum problems than... |

71 | Sur les formes quadratiques - Korkine, Zolotareff |

51 | Polynomial Time Algorithms for Finding Integer Relations among - Hastad, Just, et al. - 1989 |

49 | A knapsack-type public key cryptosystem based on arithmetic in nite elds - Chor, Rivest - 1988 |

45 |
and Lazlo Lovasz, Factoring polynomials with rational coefficients,
- Lenstra, Lenstra
- 1982
(Show Context)
Citation Context ...tor that is shortest in the sup--norm is known to be NP--complete [4] (in its feasibility recognition form). On the other hand the L 3 --lattice basis reduction algorithm of Lenstra, Lenstra, Lov'asz =-=[17]-=- is a polynomial time algorithm that finds a non--zero vector in an m--dimensional lattice that is guaranteed to be at most 2 m=2 --times the length of the shortest non-- zero vector in that lattice. ... |

32 |
Solving Low Density Knapsacks
- Brickell
- 1984
(Show Context)
Citation Context ...is reduction is needed. The knapsack or subset sum problem is to solve, given positive integers a 1 ; : : : ; a n and s, the equation n X i=1 a i x i = s with x 1 ; : : : ; x n 2 f0; 1g: The Brickell =-=[1]-=- and the Lagarias--Odlyzko [14] algorithms solve almost all subset sum problems for which the density d = n = log 2 maxa i is sufficiently small. Radziszowski and Kreher [17] evaluate the performance ... |

32 | Factoring integers and computing discrete logarithms via Diophantine approximation
- Schnorr
- 1991
(Show Context)
Citation Context ...ormance of all these algorithms in solving subset sum problems. These algorithms have also been applied to solve the diophantine approximation problem that yields the factorization of a given integer =-=[25]-=-. However to make this approach work for large integers further progress in basis reduction is needed. The knapsack or subset sum problem is to solve, given positive integers a 1 ; : : : ; a n and s, ... |

18 |
Simultaneous reduction of a lattice basis and its reciprocal basis
- Seysen
- 1993
(Show Context)
Citation Context ...uch shorter vectors than is guaranteed by the worst case 2 m=2 --bound. The performance of the L 3 has been further improved by suitable modifications [5,15,22], and new algorithms are being invented =-=[19,23,24,27]-=-. Possibly finding reasonably short vectors in a random lattice is not so difficult on the average. This would have important consequences for solving linear and non--linear integer programming proble... |

16 | Extraits de lettres de m. ch. hermite a m. jacoby sur diff~rents objects de la theorie des nombres, deuxieme lettre du 6 aofit 1845 - Hermite |

14 | Basis reduction algorithms and subset sum problems
- LaMacchia
- 1991
(Show Context)
Citation Context ... lattice. The L 3 --algorithm finds in practice much shorter vectors than is guaranteed by the worst case 2 m=2 --bound. The performance of the L 3 has been further improved by suitable modifications =-=[5,15,22]-=-, and new algorithms are being invented [19,23,24,27]. Possibly finding reasonably short vectors in a random lattice is not so difficult on the average. This would have important consequences for solv... |

7 |
Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in lattices
- van
- 1981
(Show Context)
Citation Context ...o determine the exact complexity of finding short vectors in a lattice. On the one hand the problem of finding a non--zero lattice vector that is shortest in the sup--norm is known to be NP--complete =-=[4]-=- (in its feasibility recognition form). On the other hand the L 3 --lattice basis reduction algorithm of Lenstra, Lenstra, Lov'asz [17] is a polynomial time algorithm that finds a non--zero vector in ... |

7 | Approximating integer lattices by lattices with cycle factor groups," - Paz, Schnorr - 1987 |

6 | asz. An Algorithmic Theory of Numbers, Graphs and Convexity - Lov - 1986 |

6 |
The generalized basis reduction algorithm
- asz, Scarf
- 1992
(Show Context)
Citation Context ...uch shorter vectors than is guaranteed by the worst case 2 m=2 --bound. The performance of the L 3 has been further improved by suitable modifications [5,15,22], and new algorithms are being invented =-=[19,23,24,27]-=-. Possibly finding reasonably short vectors in a random lattice is not so difficult on the average. This would have important consequences for solving linear and non--linear integer programming proble... |

6 |
Solving subset sum problems with the L algorithm
- Radziszowski, Kreher
- 1988
(Show Context)
Citation Context ... lattice. The L 3 --algorithm finds in practice much shorter vectors than is guaranteed by the worst case 2 m=2 --bound. The performance of the L 3 has been further improved by suitable modifications =-=[5,15,22]-=-, and new algorithms are being invented [19,23,24,27]. Possibly finding reasonably short vectors in a random lattice is not so difficult on the average. This would have important consequences for solv... |

5 | Block Korkin-Zolotarev bases and successive minima
- Schnorr
- 1996
(Show Context)
Citation Context ... by Hermite in his second letter to Jacobi (1845) and by Korkin and Zolotarev (1873). Theorem 3 shows the strength of Korkin--Zolotarev reduction compared to L 3 --reduction, see Theorem 1. Theorem 3 =-=[13]-=- Every Korkin--Zolotarev basis b 1 ; : : : ; b m satisfies 4 (i + 3)skb i k 2 =s2 isi + 3 4 for i = 1; : : : ; m: The fastest known algorithm for Korkin--Zolotarev reduction of a basis b 1 ; : : : ; b... |

5 |
Schnorr: A Hierarchy of Polynomial Time Lattice Basis Reduction Algorithms
- P
- 1987
(Show Context)
Citation Context ...uch shorter vectors than is guaranteed by the worst case 2 m=2 --bound. The performance of the L 3 has been further improved by suitable modifications [5,15,22], and new algorithms are being invented =-=[19,23,24,27]-=-. Possibly finding reasonably short vectors in a random lattice is not so difficult on the average. This would have important consequences for solving linear and non--linear integer programming proble... |

5 |
Lattice basis reduction: Improved algorithms and solving subset sum problems
- Schnorr, Euchner
- 1991
(Show Context)
Citation Context ... and it is made so that the values (y t + d\Gammay t c +4 t ) 2 c t do not decrease for the chosen sequence 4 t .) At this latter point t is incremented to t + 1. 4. Our original ENUM--algorithm, see =-=[26]-=-, did not enumerate the values (y t + d\Gammay t c+4 t )c t in increasing order. The new ENUM--algorithm is slightly better for block Korkin--Zolotarev reduction with pruning, see the end of section 7... |

4 |
Minkowski’s convex body theory and integer programming.
- Kannan
- 1987
(Show Context)
Citation Context ...s a theoretic worst case time bound of p n n+o(n) + O(n 4 log B) arithmetic steps on O(n log B)--bit integers [23]. This algorithm is an improved version of Kannan's shortest lattice vector algorithm =-=[11]-=-. Schnorr [23] introduced the following notion of a block Korkin--Zolotarev reduced basis. Let fi be an integer, 2sfi ! m. A lattice basis b 1 ; : : : ; b m is fi--reduced if it is size--reduced and i... |

4 | Schnorr: A More Efficient Algorithm for Lattice Basis Reduction - P - 1988 |

1 |
Praktische Algorithmen zur Gitterreduktion und Faktorisierung
- Euchner
- 1991
(Show Context)
Citation Context ... lattice. The L 3 --algorithm finds in practice much shorter vectors than is guaranteed by the worst case 2 m=2 --bound. The performance of the L 3 has been further improved by suitable modifications =-=[5,15,22]-=-, and new algorithms are being invented [19,23,24,27]. Possibly finding reasonably short vectors in a random lattice is not so difficult on the average. This would have important consequences for solv... |

1 | Frieze: On the Lagarias--Odlyzko algorithm for the subset sum problem - M - 1986 |

1 |
Improving the critical density of the Lagarias-- Odlyzko attack against subset sum problems
- Joux, Stern
- 1991
(Show Context)
Citation Context ...ersity in summer 1990. This work has been mentioned in the talk of the first author at the workshop on cryptography at Princeton University in September 1990 and has influenced the subsequent work in =-=[3,10,15]-=-. 2 Basic concepts, L 3 --reduction Let IR n be the n--dimensional real vector space with the ordinary inner product ! ; ? and Euclidean length kyk = hy; yi 1=2 . A discrete, additive subgroup L ae IR... |

1 | Odlyzko: The rise and fall of knapsack cryptosystems. Cryptology and Computational - M - 1990 |