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## A Lattice Basis Reduction Algorithm ∗

Citations: | 9 - 9 self |

### Citations

1198 |
A course in computational algebraic number theory
- Cohen
- 1993
(Show Context)
Citation Context ... algorithm finds a reduced basis, that is, a basis whose vectors are reduced in length. The lattice reduction problem arises from fields such as integer programming [2], cryptology [6], number theory =-=[4]-=-, and information theory [1]. In this paper, after a short introduction to lattices and bases in Section 2, various definitions of reduced basis are described in Section 3. They include the definition... |

331 | Closest point search in lattices
- Agrell, Eriksson, et al.
- 2002
(Show Context)
Citation Context ...asis, that is, a basis whose vectors are reduced in length. The lattice reduction problem arises from fields such as integer programming [2], cryptology [6], number theory [4], and information theory =-=[1]-=-. In this paper, after a short introduction to lattices and bases in Section 2, various definitions of reduced basis are described in Section 3. They include the definitions of Minkowski-reduced basis... |

186 |
Minkowski’s convex body theorem and integer programming
- Kannan
- 1987
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Citation Context ...ed. Step 2 is the key to the algorithm. It solves the following problem: Given a basis {a1,a2,...,an} for a lattice L and a shortest nonzero lattice point b1 = Anz, extend b1 to a new basis for L. In =-=[7]-=-, Kannan gave an algorithm for this problem. Here we present a novel unimodular matrix transformation method. A sufficient and necessary condition that b1 = Anz is extendable to a basis is that the en... |

141 |
A Course in Convexity
- Barvinok
- 2002
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Citation Context ...more” orthogonal. A basis reduction algorithm finds a reduced basis, that is, a basis whose vectors are reduced in length. The lattice reduction problem arises from fields such as integer programming =-=[2]-=-, cryptology [6], number theory [4], and information theory [1]. In this paper, after a short introduction to lattices and bases in Section 2, various definitions of reduced basis are described in Sec... |

98 |
Geometrie der Zahlen
- MINKOWSKI
- 1991
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Citation Context ...of AZ are short. We see from Figure 2 that a “better”, or “more” orthogonal, basis is shorter, or reduced in length. 3.1 Minkowski Minima The concept of reduced bases in the sense of Minkowski minima =-=[12]-=- is probably best illustrated by an example. Using the matrix B in (2) and the lattice L for which the columns of B form a basis, we consider the Euclidean length ‖Bz‖2 of a lattice point Bz. First we... |

72 | Lattice Reduction: A Toolbox for the Cryptanalyst
- Joux, Stern
- 1998
(Show Context)
Citation Context .... A basis reduction algorithm finds a reduced basis, that is, a basis whose vectors are reduced in length. The lattice reduction problem arises from fields such as integer programming [2], cryptology =-=[6]-=-, number theory [4], and information theory [1]. In this paper, after a short introduction to lattices and bases in Section 2, various definitions of reduced basis are described in Section 3. They inc... |

70 |
les formes quadratiques
- Korkine, Zolotareff, et al.
- 1873
(Show Context)
Citation Context ...e the Minkowski minima. In particular, when ω = 3/4, then η = 2 and 2 1−i λ 2 i ≤ ‖bi‖ 2 2 ≤ 2n−1 λ 2 i . 3.3 HKZ-reduced bases In the nineteenth century, Korkine and Zolotarev, see [13, Page 37] and =-=[8, 9]-=-, proposed a definition of a reduced basis by strengthening Hermite’s size-reduction. Definition 6 (HKZ-reduced) A lattice basis {b1,b2,...,bn} is called HKZ-reduced if the upper triangular matrix R i... |

61 |
Factoring polynomials with rational coefficients
- Lovász
- 1982
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Citation Context ...on (5) satisfies |ui,j| ≤ 1 , for 1 ≤ i < j ≤ n 2 |ri,j| ≤ 1 2 |ri,i|, for 1 ≤ i < j ≤ n. Often, size-reduced is a necessary condition for a reduced basis. The LLL-reduced basis is defined as follows =-=[10]-=-. The columns bi of a full column rank matrix B form an LLL-reduced basis for a lattice if the matrices Q ∗ and U in the decomposition (4) satisfy |ui,j| ≤ 1/2, j > i (size-reduced), and ‖q ∗ i ‖ 2 2 ... |

11 |
An improved LLL algorithm
- Luk, Tracy
- 2008
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Citation Context ...j > i (size-reduced), and ‖q ∗ i ‖ 2 2 + u 2 i−1,i‖q ∗ i−1‖ 2 2 ≥ ω‖q ∗ i−1‖ 2 2, where 1/4 < ω < 1. In terms of the QR decomposition (5), using the relations in (6), we have the following definition =-=[11]-=-. Definition 5 (LLL-Reduced) Given an ω ∈ (0.25,1.0), a lattice basis {b1,b2,...,bn} is called LLL-reduced if the upper triangular matrix R in the decomposition (5) of B = [b1 b2 ... bn] satisfies |ri... |

6 | An Introduction to the Geometry of Numbers, Second Printing - Cassels - 1997 |

3 |
Sur les formes quadratiques positives ternaires
- Korkine, Zolotareff
(Show Context)
Citation Context ...e the Minkowski minima. In particular, when ω = 3/4, then η = 2 and 2 1−i λ 2 i ≤ ‖bi‖ 2 2 ≤ 2n−1 λ 2 i . 3.3 HKZ-reduced bases In the nineteenth century, Korkine and Zolotarev, see [13, Page 37] and =-=[8, 9]-=-, proposed a definition of a reduced basis by strengthening Hermite’s size-reduction. Definition 6 (HKZ-reduced) A lattice basis {b1,b2,...,bn} is called HKZ-reduced if the upper triangular matrix R i... |