A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
Polynomial time algorithms for finding integer relations among real numbers
SIAM Journal on Computing
Polynomial threshold functions, AC0 functions, and spectral norms
SIAM Journal on Computing
Learning decision trees using the Fourier spectrum
SIAM Journal on Computing
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
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
On the limits of nonapproximability of lattice problems
Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
A note on the non-NP-hardness of approximate lattice problems under general Cook reductions
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
Exponential lower bound for 2-query locally decodable codes via a quantum argument
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Improved algorithms for integer programming and related lattice problems
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
The Inapproximability of Lattice and Coding Problems with Preprocessing
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
A Lattice Problem in Quantum NP
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Lower bounds for local search by quantum arguments
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Hardness of Approximating the Shortest Vector Problem in Lattices
FOCS '04 Proceedings of the 45th 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
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
Foundations and Trends® in Theoretical Computer Science
If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances
Computational Complexity
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
Noninteractive Statistical Zero-Knowledge Proofs for Lattice Problems
CRYPTO 2008 Proceedings of the 28th Annual conference on Cryptology: Advances in Cryptology
Cryptographic hardness for learning intersections of halfspaces
Journal of Computer and System Sciences
Explicit Hard Instances of the Shortest Vector Problem
PQCrypto '08 Proceedings of the 2nd International Workshop on Post-Quantum Cryptography
On lattices, learning with errors, random linear codes, and cryptography
Journal of the ACM (JACM)
Asymptotically efficient lattice-based digital signatures
TCC'08 Proceedings of the 5th conference on Theory of cryptography
A full characterization of quantum advice
Proceedings of the forty-second ACM symposium on Theory of computing
Public-key cryptography from different assumptions
Proceedings of the forty-second ACM symposium on Theory of computing
Proceedings of the forty-second ACM symposium on Theory of computing
Postselection finite quantum automata
UC'10 Proceedings of the 9th international conference on Unconventional computation
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
Towards non-black-box lower bounds in cryptography
TCC'11 Proceedings of the 8th conference on Theory of cryptography
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Generalized compact knapsacks are collision resistant
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part II
Lattice mixing and vanishing trapdoors: a framework for fully secure short signatures and more
PKC'10 Proceedings of the 13th international conference on Practice and Theory in Public Key Cryptography
Lattice enumeration using extreme pruning
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
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
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
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
A complexity analysis of a Jacobi method for lattice basis reduction
Proceedings of the Fifth International C* Conference on Computer Science and Software Engineering
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
ACM Transactions on Computation Theory (TOCT) - Special issue on innovations in theoretical computer science 2012
Hi-index | 0.00 |
We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of &nradic; lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk [1993], Goldreich and Goldwasser [2000], and Aharonov and Regev [2003]. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier series over the lattice. This technique might be useful elsewhere---we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev[2003]. An interesting fact is that our result emerged from a “dequantization” of our previous quantum result in Aharonov and Regev [2003]. This route to proving purely classical results might be beneficial elsewhere.