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Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
A hierarchy of polynomial time lattice basis reduction algorithms
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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
A public-key cryptosystem with worst-case/average-case equivalence
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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
Approximating shortest lattice vectors is not harder than approximating closet lattice vectors
Information Processing Letters
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
A new transference theorem in the geometry of numbers and new bounds for Ajtai's connection factor
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New lattice based cryptographic constructions
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An Improved Worst-Case to Average-Case Connection for Lattice Problems
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Approximating-CVP to within Almost-Polynomial Factors is NP-Hard
FOCS '98 Proceedings of the 39th 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
Improved algorithms for integer programming and related lattice problems
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
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
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
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
Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions
Computational Complexity
Efficient reductions among lattice problems
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Limits on the Hardness of Lattice Problems in lp Norms
Computational Complexity
Cryptography Based on Quadratic Forms: Complexity Considerations
Research in Cryptology
Sampling methods for shortest vectors, closest vectors and successive minima
Theoretical Computer Science
List decoding tensor products and interleaved codes
Proceedings of the forty-first annual ACM symposium on Theory of computing
A deterministic reduction for the gap minimum distance problem: [extended abstract]
Proceedings of the forty-first annual ACM symposium on Theory of computing
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The geometry of lattice cryptography
Foundations of security analysis and design VI
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
List Decoding Tensor Products and Interleaved Codes
SIAM Journal on Computing
Parallel shortest lattice vector enumeration on graphics cards
AFRICACRYPT'10 Proceedings of the Third international conference on Cryptology in Africa
Public key identification based on the equivalence of quadratic forms
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
Sampling methods for shortest vectors, closest vectors and successive minima
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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Let p 1 be any fixed real. We show that assuming NP ⊈ RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in ℓp norm within a constant factor. Under the stronger assumption NP ⊈ RTIME(2poly(log n)), we show that there is no polynomial-time algorithm with approximation ratio 2(log n)1/2−ε where n is the dimension of the lattice and ε 0 is an arbitrarily small constant.We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product, a new variant of tensor product that we introduce. This enables us to boost the hardness factor to 2(log n)1/2-ε.