Solving low-density subset sum problems
Journal of the ACM (JACM)
Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
Generating hard instances of lattice problems (extended abstract)
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
A public-key cryptosystem with worst-case/average-case equivalence
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
SIAM Journal on 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 the SVP to within a factor (1+1/dimE) is NP-Hard under randomized reductions
Journal of Computer and System Sciences
On the limits of nonapproximability of lattice problems
Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
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
Better algorithms for high-dimensional proximity problems via asymmetric embeddings
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Stable distributions, pseudorandom generators, embeddings and data stream computation
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Hardness of Approximating the Shortest Vector Problem in High Lp Norms
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
The inapproximability of lattice and coding problems with preprocessing
Journal of Computer and System Sciences - Special issue on computational complexity 2002
Hardness of Approximating the Shortest Vector Problem in Lattices
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
New lattice-based cryptographic constructions
Journal of the ACM (JACM)
Euclidean distortion and the sparsest cut
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Journal of the ACM (JACM)
The hardness of the closest vector problem with preprocessing
IEEE Transactions on Information Theory
Improved inapproximability of lattice and coding problems with preprocessing
IEEE Transactions on Information Theory
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
Lattices that admit logarithmic worst-case to average-case connection factors
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Uncertainty principles, extractors, and explicit embeddings of l2 into l1
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Foundations and Trends® in Theoretical Computer Science
Limits on the Hardness of Lattice Problems in lp Norms
Computational Complexity
Efficient lattice-based signature scheme
International Journal of Applied Cryptography
Sampling methods for shortest vectors, closest vectors and successive minima
Theoretical Computer Science
A digital signature scheme based on CV P∞
PKC'08 Proceedings of the Practice and theory in public key cryptography, 11th international conference on Public key cryptography
On bounded distance decoding for general lattices
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Parallel shortest lattice vector enumeration on graphics cards
AFRICACRYPT'10 Proceedings of the Third international conference on Cryptology in Africa
2log1-ε n hardness for the closest vector problem with preprocessing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Theoretical Computer Science
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We present reductions from lattice problems in the l2 norm to the corresponding problems in other norms such as l1, l∞ (and in fact in any other lp norm where 1 ≤ p ≤ ∞). We consider lattice problems such as the Shortest Vector Problem, Shortest Independent Vector Problem, Closest Vector Problem and the Closest Vector Problem with Preprocessing. Most reductions are simple and follow from known constructions of embeddings of normed spaces.Among other things, our reductions imply that the Shortest Vector Problem in the l1 norm and the Closest Vector Problem with Preprocessing in the l∞ norm are hard to approximate to within any constant (and beyond). Previously, the former problem was known to be hard to approximate to within 2-ε, while no hardness result was known for the latter problem.