Improved low-density subset sum algorithms
Computational Complexity
Generating hard instances of lattice problems (extended abstract)
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
The hardness of approximate optima in lattices, codes, and systems of linear equations
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
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
Some optimal inapproximability results
Journal of the ACM (JACM)
Handbook of Coding Theory
Complexity of Lattice Problems
Complexity of Lattice Problems
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant
SIAM Journal on Computing
Approximation Hardness of Bounded Degree MIN-CSP and MIN-BISECTION
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
An Improved Worst-Case to Average-Case Connection for Lattice Problems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Hardness of Approximating the Minimum Distance of a Linear Code
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
The intractability of computing the minimum distance of a code
IEEE Transactions on Information Theory
The hardness of the closest vector problem with preprocessing
IEEE Transactions on Information Theory
Maximum-likelihood decoding of Reed-Solomon codes is NP-hard
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Hardness of Approximating the Closest Vector Problem with Pre-Processing
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Lattice problems and norm embeddings
Proceedings of the thirty-eighth 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
Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions
Computational Complexity
Limits on the Hardness of Lattice Problems in lp Norms
Computational Complexity
Proceedings of the forty-second ACM symposium on Theory of computing
The Euclidean distortion of flat tori
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Tight approximability results for the maximum solution equation problem over Zp
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
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
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 prove that the closest vector problem with preprocessing (CVPP) is NP-hard to approximate within any factor less than √5/3. More specifically, we show that there exists a reduction from an NP-hard problem to the approximate closest vector problem such that the lattice depends only on the size of the original problem, and the specific instance is encoded solely in the target vector. It follows that there are lattices for which the closest vector problem cannot be approximated within factors γ lp, norm, for p ≥ 1, showing that CVPP in the lp norm is hard to approximate within any factor γ p√5/3. As an intermediate step, we establish analogous results for the nearest codeword problem with preprocessing (NCPP), proving that for any finite field GF(q), NCPP over GF(q) is NP-hard to approximate within any factor less than 5/3.