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## A Delayed Size-reduction Technique for Speeding Up the LLL Algorithm

Citations: | 1 - 1 self |

### Citations

960 | Factoring polynomials with rational coefficients
- Lenstra, Lenstra, et al.
- 1982
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Citation Context ...ur algorithm provides a starting point for parallel LLL algorithms. Keywords Lattice, lattice basis reduction, LLL algorithm. 1 Introduction The LLL algorithm authored by Lenstra, Lenstra, and Lovász =-=[11]-=- is a lattice basis reduction method. The problem of reducing a lattice basis has wide applications: factorization of polynomials with rational coefficients [11], cryptography [3], integer programming... |

331 | Closest point search in lattices
- Agrell, Eriksson, et al.
- 2002
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Citation Context ...educing a lattice basis has wide applications: factorization of polynomials with rational coefficients [11], cryptography [3], integer programming [12], number theory [12], and digital communications =-=[1, 9]-=-, just to name a few. Although it does not guarantee a best reduced lattice basis, the LLL algorithm can efficiently produce reasonably good results. Thus the LLL algorithm is widely used in wireless ... |

271 | On maximum likelihood detection and the search for the closest lattice point
- Damen, Gamal, et al.
- 2003
(Show Context)
Citation Context ...ciently produce reasonably good results. Thus the LLL algorithm is widely used in wireless communications to improve the performance. For example, it is used in lattice-reduction-aided MIMO detection =-=[4, 5, 6, 16, 18]-=-, where lattice basis reduction is used as a preprocessing. In time critical situations such as delay-constrained MIMO detection [8], speed is crucial. Thus the LLL algorithm has been modified to impr... |

235 |
solutions to polynomial equations, and low exponent RSA vulnerabilities”, Journal of Cryptology 10, 233–260; see also
- Coppersmith, “Small
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Citation Context ...Lenstra, and Lovász [11] is a lattice basis reduction method. The problem of reducing a lattice basis has wide applications: factorization of polynomials with rational coefficients [11], cryptography =-=[3]-=-, integer programming [12], number theory [12], and digital communications [1, 9], just to name a few. Although it does not guarantee a best reduced lattice basis, the LLL algorithm can efficiently pr... |

124 |
An Algorithmic Theory of Numbers, Graphs, and Convexity.
- Lovasz
- 1986
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Citation Context ...is a lattice basis reduction method. The problem of reducing a lattice basis has wide applications: factorization of polynomials with rational coefficients [11], cryptography [3], integer programming =-=[12]-=-, number theory [12], and digital communications [1, 9], just to name a few. Although it does not guarantee a best reduced lattice basis, the LLL algorithm can efficiently produce reasonably good resu... |

109 | Lattice-reduction-aided detectors for MIMO communication systems
- Yao, Wornell
- 2002
(Show Context)
Citation Context ...ciently produce reasonably good results. Thus the LLL algorithm is widely used in wireless communications to improve the performance. For example, it is used in lattice-reduction-aided MIMO detection =-=[4, 5, 6, 16, 18]-=-, where lattice basis reduction is used as a preprocessing. In time critical situations such as delay-constrained MIMO detection [8], speed is crucial. Thus the LLL algorithm has been modified to impr... |

70 | Efficient algorithm for decoding layered space-time codes
- Wübben, Böhnke, et al.
- 2001
(Show Context)
Citation Context ...ity) and NSERC of Canada. 1a 2 b 1 b 2 a 1 Figure 1: The columns of A (1) and B (2), two bases for the lattice L. speed without sacrificing the quality of the reduced basis produced by the algorithm =-=[7, 8, 17]-=-. In this paper, we propose a technique, called delayed size-reduction, to speed up the LLL algorithm. The modified algorithm can significantly speed up the LLL algorithm without sacrificing the quali... |

69 | Near-maximum-likelihood detection of MIMO systems using MMSE-based lattice-reduction
- Wübben, Böhnke, et al.
- 2004
(Show Context)
Citation Context ...ciently produce reasonably good results. Thus the LLL algorithm is widely used in wireless communications to improve the performance. For example, it is used in lattice-reduction-aided MIMO detection =-=[4, 5, 6, 16, 18]-=-, where lattice basis reduction is used as a preprocessing. In time critical situations such as delay-constrained MIMO detection [8], speed is crucial. Thus the LLL algorithm has been modified to impr... |

58 | Complex lattice reduction algorithm for low-complexity full-diversity MIMO detection
- Gan, Ling, et al.
- 2009
(Show Context)
Citation Context ...ity) and NSERC of Canada. 1a 2 b 1 b 2 a 1 Figure 1: The columns of A (1) and B (2), two bases for the lattice L. speed without sacrificing the quality of the reduced basis produced by the algorithm =-=[7, 8, 17]-=-. In this paper, we propose a technique, called delayed size-reduction, to speed up the LLL algorithm. The modified algorithm can significantly speed up the LLL algorithm without sacrificing the quali... |

11 |
An improved LLL algorithm
- Luk, Tracy
- 2008
(Show Context)
Citation Context ...n (4), the column lengths are expected to be shortened. When A is an integer or rational matrix, the LLL algorithm can be performed in exact arithmetic in polynomial time [11]. In 2008, Luk and Tracy =-=[14]-=- presented a floating-point version of the algorithm for real lattice basis matrices. In this paper, we consider the real case and adopt the floating-point version in [14]. However, the technique prop... |

7 |
Novel joint sorting and reduction technique for delay-constrained LLL-aided MIMO detection
- Gan, Mow
- 2008
(Show Context)
Citation Context ... it is used in lattice-reduction-aided MIMO detection [4, 5, 6, 16, 18], where lattice basis reduction is used as a preprocessing. In time critical situations such as delay-constrained MIMO detection =-=[8]-=-, speed is crucial. Thus the LLL algorithm has been modified to improve its ∗ Wen Zhang is partially supported by the National Natural Science Foundation of China under grant 10871051, Doctoral Progra... |

6 |
An Introduction to the Geometry of Numbers, Second Printing
- Cassels
- 1997
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Citation Context ...We say that {b1, b2} is a reduced basis for L. Given a basis for a lattice, a lattice reduction method produces a reduced basis for the lattice. For details of lattices, bases, and reduced bases, see =-=[2, 10]-=-. 23 The LLL Algorithm The LLL algorithm is a lattice basis reduction method. It consists of two stages. Given an m-by-n lattice generator matrix A, the first stage triangularizes it using the Gram-S... |

1 |
An efficient algorithm for clustered integer least squares problems
- Chun, Park
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1 | A parallel LLL algorithm - Luo, Qiao - 2010 |

1 |
The LLL Algorithm: Survey and Applications. Information Security and Cryptography, Texts and Monographs
- Vallée
- 2010
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Citation Context ...ameter that controls the rate of convergence of the algorithm. The condition (4) requires small off-diagonal elements, thus short column lengths. This condition is often called size-reduced condition =-=[15]-=-. The second condition (5) imposes a loosely increasing order on the diagonal elements di. By pushing up small diagonal elements and enforcing the size-reduced condition (4), the column lengths are ex... |