Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
A course in computational algebraic number theory
A course in computational algebraic number theory
A public-key cryptosystem with worst-case/average-case equivalence
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
The shortest vector problem in L2 is NP-hard for randomized reductions (extended abstract)
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
On the complexity of computing short linearly independent vectors and short bases in a lattice
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Approximating shortest lattice vectors is not harder than approximating closet lattice vectors
Information Processing Letters
A sieve algorithm for the shortest lattice vector problem
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Complexity of Lattice Problems
Complexity of Lattice Problems
The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant
SIAM Journal on Computing
A new transference theorem in the geometry of numbers and new bounds for Ajtai's connection factor
Discrete Applied Mathematics - Special issue: Special issue devoted to the fifth annual international computing and combinatories conference (COCOON'99) Tokyo, Japan 26-28 July 1999
Public-Key Cryptosystems from Lattice Reduction Problems
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
Sampling Short Lattice Vectors and the Closest Lattice Vector Problem
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
Almost Perfect Lattices, the Covering Radius Problem, and Applications to Ajtai's Connection Factor
SIAM Journal on Computing
New lattice-based cryptographic constructions
Journal of the ACM (JACM)
Hardness of approximating the shortest vector problem in lattices
Journal of the ACM (JACM)
Tensor-based hardness of the shortest vector problem to within almost polynomial factors
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Worst-Case to Average-Case Reductions Based on Gaussian Measures
SIAM Journal on Computing
Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions
Computational Complexity
The complexity of the covering radius problem
Computational Complexity
Sampling methods for shortest vectors, closest vectors and successive minima
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Sampling methods for shortest vectors, closest vectors and successive minima
Theoretical Computer Science
Lattice-based identification schemes secure under active attacks
PKC'08 Proceedings of the Practice and theory in public key cryptography, 11th international conference on Public key cryptography
Proceedings of the forty-second ACM symposium on Theory of computing
Algorithms for the shortest and closest lattice vector problems
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
Approximating the closest vector problem using an approximate shortest vector oracle
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
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We give various deterministic polynomial time reductions among approximation problems on point lattices. Our reductions are both efficient and robust, in the sense that they preserve the rank of the lattice and approximation factor achieved. Our main result shows that for any γ ≥ 1, approximating all the successive minima of a lattice (and, in particular, approximately solving the Shortest Independent Vectors Problem, SIVPγ) within a factor γ reduces under deterministic polynomial time rank-preserving reductions to approximating the Closest Vector Problem (CVP) within the same factor γ. This solves an open problem posed by Blömer in (ICALP 2000). As an application, we obtain faster algorithms for the exact solution of SIVP that run in time n! · sO(1) (where n is the rank of the lattice, and s the size of the input,) improving on the best previously known solution of Blömer (ICALP 2000) by a factor 3n. We also show that SIVP, CVP and many other lattice problems are equivalent in their exact version under deterministic polynomial time rank-preserving reductions.