## An Improved Worst-Case to Average-Case Connection for Lattice Problems (extended abstract) (1997)

Venue: | In FOCS |

Citations: | 56 - 11 self |

### BibTeX

@INPROCEEDINGS{Cai97animproved,

author = {Jin-yi Cai and Ajay P. Nerurkar},

title = {An Improved Worst-Case to Average-Case Connection for Lattice Problems (extended abstract)},

booktitle = {In FOCS},

year = {1997},

pages = {468--477},

publisher = {IEEE}

}

### Years of Citing Articles

### OpenURL

### Abstract

We improve a connection of the worst-case complexity and the average-case complexity of some well-known lattice problems. This fascinating connection was first discovered by Ajtai [1] in 1996. We improve the exponent of this connection from 8 to 3:5 + ffl. Department of Computer Science, State University of New York at Buffalo, Buffalo, NY 14260. Research supported in part by NSF grants CCR-9319393 and CCR-9634665, and an Alfred P. Sloan Fellowship. Email: cai@cs.buffalo.edu y Department of Computer Science, State University of New York at Buffalo, Buffalo, NY 14260. Research supported in part by NSF grants CCR-9319393 and CCR-9634665. Email: apn@cs.buffalo.edu 1 Introduction A lattice L is a discrete additive subgroup of R n . There are many fascinating problems concerning lattices, both from a structural and from an algorithmic point of view [12, 20, 11, 13]. The study of lattice problems can be traced back to Gauss, Dirichlet and Hermite, among others [8, 6, 14]. The subje...

### Citations

1145 |
A.: Geometric Algorithms and Combinatorial Optimization
- Grötschel, Lovász, et al.
- 1988
(Show Context)
Citation Context ...s.buffalo.edu 1 Introduction A lattice L is a discrete additive subgroup of R n . There are many fascinating problems concerning lattices, both from a structural and from an algorithmic point of view =-=[12, 20, 11, 13]-=-. The study of lattice problems can be traced back to Gauss, Dirichlet and Hermite, among others [8, 6, 14]. The subject was first conceived as a bridge between geometry and Diophantine approximation ... |

703 | Factoring polynomials with rational coefficients
- Lenstra, Lenstra, et al.
- 1982
(Show Context)
Citation Context ...his fundamental theorems on shortest vectors and successive minima. In recent years, there is enormous interest in the algorithmic aspects of the theory, especially in connection with basis reduction =-=[18, 23]-=-, algorithmic Diophantine approximation and combinatorial optimization [11], integer programming [19], volume estimation for convex bodies [7, 21, 15] and, cryptography [1, 2, 10, 17]. There is an inh... |

235 | Integer programming with a fixed number of variables
- Lenstra
- 1983
(Show Context)
Citation Context ...terest in the algorithmic aspects of the theory, especially in connection with basis reduction [18, 23], algorithmic Diophantine approximation and combinatorial optimization [11], integer programming =-=[19]-=-, volume estimation for convex bodies [7, 21, 15] and, cryptography [1, 2, 10, 17]. There is an inherent beauty in many problems in the theory of Geometry of Numbers. Moreover, major algorithmic progr... |

233 |
On Lovász’ lattice reduction and the nearest lattice point problem
- Babai
- 1986
(Show Context)
Citation Context ...chnorr's algorithm gets a bound of (1 + ffl) n , but the running time badly depends on ffl in the exponent [23]. Babai gave an algorithm that approximates the nearest vector by a factor of (3= p 2) n =-=[4]-=-. The recent breakthrough by Ajtai [1] has its motivations from cryptography, and the connection between average-case and worst-case complexity in general. It has been realized for some time that the ... |

206 | A Public-Key Cryptosystem with Worst-Case/Average-Case Equivalence
- Ajtai, Dwork
(Show Context)
Citation Context ...ear combination of remainder vectors. Moreover, Ball's theorem [5] implies that one can get independent short lattice vectors by repeating this process. Based on the reduction in [1], Ajtai and Dwork =-=[2]-=- have proposed a public-key cryptosystem with provable security guarantees based on worst-case hardness assumption. Another public-key system based on lattice problems was proposed in [10], although n... |

165 |
Generating hard instances of lattice problems
- Ajtai
- 2004
(Show Context)
Citation Context ...Nerurkar y Abstract We improve a connection of the worst-case complexity and the average-case complexity of some well-known lattice problems. This fascinating connection was first discovered by Ajtai =-=[1]-=- in 1996. We improve the exponent of this connection from 8 to 3:5 + ffl. Department of Computer Science, State University of New York at Buffalo, Buffalo, NY 14260. Research supported in part by NSF ... |

154 | The hardn-8 of approximate optima in lattices, codes, and systems of linear equations
- Arora, Babai, et al.
- 1997
(Show Context)
Citation Context ...em is NPhard for the l 1 -norm, but it is not known whether it is NP-hard under any other l p norm. Van Emde Boas [24] showed that finding the nearest vector is NP-hard under all l p norms, ps1. From =-=[3]-=- it is known that finding an approximate solution to within any constant factor for the nearest vector problem for any l p norm, and, for the shortest vector problem in the l 1 -norm, are both NP-hard... |

147 | A random polynomial time algorithm for approximating the volume of convex bodies
- Dyer, Kannan
- 1991
(Show Context)
Citation Context ...heory, especially in connection with basis reduction [18, 23], algorithmic Diophantine approximation and combinatorial optimization [11], integer programming [19], volume estimation for convex bodies =-=[7, 21, 15]-=- and, cryptography [1, 2, 10, 17]. There is an inherent beauty in many problems in the theory of Geometry of Numbers. Moreover, major algorithmic progress in the field, such as Lov'asz's basis reducti... |

123 |
Geometry of Numbers
- Gruber, Lekkerkerker
- 1987
(Show Context)
Citation Context ...s.buffalo.edu 1 Introduction A lattice L is a discrete additive subgroup of R n . There are many fascinating problems concerning lattices, both from a structural and from an algorithmic point of view =-=[12, 20, 11, 13]-=-. The study of lattice problems can be traced back to Gauss, Dirichlet and Hermite, among others [8, 6, 14]. The subject was first conceived as a bridge between geometry and Diophantine approximation ... |

122 | Public-key cryptosystems from lattice reduction problems
- Goldreich, Goldwasser, et al.
(Show Context)
Citation Context ...n with basis reduction [18, 23], algorithmic Diophantine approximation and combinatorial optimization [11], integer programming [19], volume estimation for convex bodies [7, 21, 15] and, cryptography =-=[1, 2, 10, 17]-=-. There is an inherent beauty in many problems in the theory of Geometry of Numbers. Moreover, major algorithmic progress in the field, such as Lov'asz's basis reduction algorithm, has had a tremendou... |

107 | A.M.: Solving low-density subset sum problems
- Lagarias, Odlyzko
- 1985
(Show Context)
Citation Context ...n with basis reduction [18, 23], algorithmic Diophantine approximation and combinatorial optimization [11], integer programming [19], volume estimation for convex bodies [7, 21, 15] and, cryptography =-=[1, 2, 10, 17]-=-. There is an inherent beauty in many problems in the theory of Geometry of Numbers. Moreover, major algorithmic progress in the field, such as Lov'asz's basis reduction algorithm, has had a tremendou... |

101 |
An Algorithmic Theory of Numbers, Graphs and Convexity
- Lovász
- 1986
(Show Context)
Citation Context ...s.buffalo.edu 1 Introduction A lattice L is a discrete additive subgroup of R n . There are many fascinating problems concerning lattices, both from a structural and from an algorithmic point of view =-=[12, 20, 11, 13]-=-. The study of lattice problems can be traced back to Gauss, Dirichlet and Hermite, among others [8, 6, 14]. The subject was first conceived as a bridge between geometry and Diophantine approximation ... |

69 | Simonovits K. Isoperimetric problems for convex bodies and a localization lemma
- Kannan, Lovász
(Show Context)
Citation Context ...heory, especially in connection with basis reduction [18, 23], algorithmic Diophantine approximation and combinatorial optimization [11], integer programming [19], volume estimation for convex bodies =-=[7, 21, 15]-=- and, cryptography [1, 2, 10, 17]. There is an inherent beauty in many problems in the theory of Geometry of Numbers. Moreover, major algorithmic progress in the field, such as Lov'asz's basis reducti... |

66 |
The mixing rate of markov chains, an isoperimetric inequality, and computing the volume
- Lovász, Simonovits
- 1990
(Show Context)
Citation Context ...heory, especially in connection with basis reduction [18, 23], algorithmic Diophantine approximation and combinatorial optimization [11], integer programming [19], volume estimation for convex bodies =-=[7, 21, 15]-=- and, cryptography [1, 2, 10, 17]. There is an inherent beauty in many problems in the theory of Geometry of Numbers. Moreover, major algorithmic progress in the field, such as Lov'asz's basis reducti... |

57 |
The computational complexity of simultaneous diophantine approximation problems
- Lagarias
- 1985
(Show Context)
Citation Context ...derlying so much fascination and activity is the belief, yet not a proof, that many of the well-known algorithmic problems for lattices are computationally hard for P. Regarding NP-hardness, Lagarias =-=[16]-=- showed that the shortest vector problem is NPhard for the l 1 -norm, but it is not known whether it is NP-hard under any other l p norm. Van Emde Boas [24] showed that finding the nearest vector is N... |

56 | Collision-free hashing from lattice problems. Available from ECCC as - Goldreich, Goldwasser, et al. |

52 |
Another NP-complete partition problem and the complexity of computing short vectors in a lattice
- Boas
- 1981
(Show Context)
Citation Context ...hard for P. Regarding NP-hardness, Lagarias [16] showed that the shortest vector problem is NPhard for the l 1 -norm, but it is not known whether it is NP-hard under any other l p norm. Van Emde Boas =-=[24]-=- showed that finding the nearest vector is NP-hard under all l p norms, ps1. From [3] it is known that finding an approximate solution to within any constant factor for the nearest vector problem for ... |

32 |
Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. Journal für die Reine und Angewandte Mathematik
- Dirichlet
(Show Context)
Citation Context ...lems concerning lattices, both from a structural and from an algorithmic point of view [12, 20, 11, 13]. The study of lattice problems can be traced back to Gauss, Dirichlet and Hermite, among others =-=[8, 6, 14]-=-. The subject was first conceived as a bridge between geometry and Diophantine approximation and the theory of quadratic forms. The field Geometry of Numbers was christened by Minkowski when he proved... |

30 |
A hierarchy of polynomial time basis reduction algorithms
- Schnorr
- 1987
(Show Context)
Citation Context ...his fundamental theorems on shortest vectors and successive minima. In recent years, there is enormous interest in the algorithmic aspects of the theory, especially in connection with basis reduction =-=[18, 23]-=-, algorithmic Diophantine approximation and combinatorial optimization [11], integer programming [19], volume estimation for convex bodies [7, 21, 15] and, cryptography [1, 2, 10, 17]. There is an inh... |

21 |
Cube slicing in
- Ball
- 1986
(Show Context)
Citation Context ...udo-cube has roughly the same number of lattice points. The proof relies heavily on the geometric properties of the set-up, in terms of eigenvalues and singular values. A recent theorem of Keith Ball =-=[5]-=-, which gives a precise upper bound on the volume of the intersection of any hyperplane with the unit cube, also plays a role in the proof. Once we achieved a reasonable level of uniformity in the num... |

15 |
Extraits de lettres de M. Ch. Hermite à M. Jacobi sur differénts objets de la théorie de nombres, (Crelle’s) Journal für die Reine und Angewandte Mathematik
- Hermite
(Show Context)
Citation Context ...lems concerning lattices, both from a structural and from an algorithmic point of view [12, 20, 11, 13]. The study of lattice problems can be traced back to Gauss, Dirichlet and Hermite, among others =-=[8, 6, 14]-=-. The subject was first conceived as a bridge between geometry and Diophantine approximation and the theory of quadratic forms. The field Geometry of Numbers was christened by Minkowski when he proved... |

7 |
te Riele - Disproof of the Mertens conjecture, Journ. reine angew. Mathematik 357
- Odlyzko, H
- 1985
(Show Context)
Citation Context ...thmic progress in the field, such as Lov'asz's basis reduction algorithm, has had a tremendous impact on many other subjects (e.g., integer programming [19], or the disproof of the Mertens conjecture =-=[22]-=-). However, underlying so much fascination and activity is the belief, yet not a proof, that many of the well-known algorithmic problems for lattices are computationally hard for P. Regarding NP-hardn... |