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
Algorithmic complexity in coding theory and the minimum distance problem
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Finding the closest lattice vector when it's unusually close
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
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
Sampling Short Lattice Vectors and the Closest Lattice Vector Problem
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
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
Journal of the ACM (JACM)
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
On the complexity of decoding lattices using the Korkin-Zolotarev reduced basis
IEEE Transactions on Information Theory
The hardness of the closest vector problem with preprocessing
IEEE Transactions on Information Theory
Closest point search in lattices
IEEE Transactions on Information Theory
Hardness of approximating the minimum distance of a linear code
IEEE Transactions on Information Theory
Improved inapproximability of lattice and coding problems with preprocessing
IEEE Transactions on Information Theory
Trapdoors for hard lattices and new cryptographic constructions
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Limits on the Hardness of Lattice Problems in lp Norms
Computational Complexity
Simultaneous Hardcore Bits and Cryptography against Memory Attacks
TCC '09 Proceedings of the 6th Theory of Cryptography Conference on Theory of Cryptography
On lattices, learning with errors, random linear codes, and cryptography
Journal of the ACM (JACM)
Algorithms for the shortest and closest lattice vector problems
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
The geometry of lattice cryptography
Foundations of security analysis and design VI
A note on BDD problems with λ2 -gap
Information Processing Letters
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A central problem in the algorithmic study of lattices is the closest vector problem: given a lattice $\mathcal{L}$ represented by some basis, and a target point $\vec{y}$, find the lattice point closest to $\vec{y}$. Bounded Distance Decoding is a variant of this problem in which the target is guaranteed to be close to the lattice, relative to the minimum distance $\lambda_1(\mathcal{L})$ of the lattice. Specifically, in the α-Bounded Distance Decoding problem (α-BDD), we are given a lattice $\mathcal{L}$ and a vector $\vec{y}$ (within distance $\alpha\cdot\lambda_1(\mathcal{L})$ from the lattice), and we are asked to find a lattice point $\vec{x}\in \mathcal{L}$ within distance $\alpha\cdot\lambda_1(\mathcal{L})$ from the target. In coding theory, the lattice points correspond to codewords, and the target points correspond to lattice points being perturbed by noise vectors. Since in coding theory the lattice is usually fixed, we may “pre-process” it before receiving any targets, to make the subsequent decoding faster. This leads us to consider α-BDD with pre-processing. We show how a recent technique of Aharonov and Regev [2] can be used to solve α-BDD with pre-processing in polynomial time for $\alpha=O\left(\sqrt{(\log{n})/n}\right)$. This improves upon the previously best known algorithm due to Klein [13] which solved the problem for $\alpha=O\left(1/n\right)$. We also establish hardness results for α-BDD and α-BDD with pre-processing, as well as generalize our results to other ℓp norms.