Algorithms to construct Minkowski reduced and Hermite reduced lattice bases
Theoretical Computer Science
Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
A more efficient algorithm for lattice basis reduction
Journal of Algorithms
Improved low-density subset sum algorithms
Computational Complexity
Lattice basis reduction: improved practical algorithms and solving subset sum problems
Mathematical Programming: Series A and B
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
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 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
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
Solvability by radicals is in polynomial time
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of 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
Almost Perfect Lattices, the Covering Radius Problem, and Applications to Ajtai's Connection Factor
SIAM Journal on Computing
Journal of the ACM (JACM)
Hardness of approximating the shortest vector problem in lattices
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
Hardness of the Covering Radius Problem on Lattices
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
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
Journal of Computer and System Sciences
Worst-Case to Average-Case Reductions Based on Gaussian Measures
SIAM Journal on Computing
Efficient reductions among lattice problems
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Finding short lattice vectors within mordell's inequality
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The complexity of the covering radius problem
Computational Complexity
Sampling methods for shortest vectors, closest vectors and successive minima
Theoretical Computer Science
Finding the Closest Lattice Point by Iterative Slicing
SIAM Journal on Discrete Mathematics
Fast LLL-type lattice reduction
Information and Computation
Improved analysis of Kannan's shortest lattice vector algorithm
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Faster exponential time algorithms for the shortest vector problem
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Rankin's constant and blockwise lattice reduction
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
Computing the Voronoi cell of a lattice: the diamond-cutting algorithm
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
Improved inapproximability of lattice and coding problems with preprocessing
IEEE Transactions on Information Theory
The Hardness of the Closest Vector Problem With Preprocessing Over Norm
IEEE Transactions on Information Theory
On the complexity of circuit satisfiability
Proceedings of the forty-second ACM symposium on Theory of computing
Accelerating lattice reduction with FPGAs
LATINCRYPT'10 Proceedings of the First international conference on Progress in cryptology: cryptology and information security in Latin America
Adaptively secure identity-based identification from lattices without random oracles
SCN'10 Proceedings of the 7th international conference on Security and cryptography for networks
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
Better key sizes (and attacks) for LWE-based encryption
CT-RSA'11 Proceedings of the 11th international conference on Topics in cryptology: CT-RSA 2011
Analysis of gauss-sieve for solving the shortest vector problem in lattices
WALCOM'11 Proceedings of the 5th international conference on WALCOM: algorithms and computation
Improved Nguyen-Vidick heuristic sieve algorithm for shortest vector problem
Proceedings of the 6th ACM Symposium on Information, Computer and Communications Security
Covering cubes and the closest vector problem
Proceedings of the twenty-seventh annual symposium on Computational geometry
Lattice reduction algorithms: theory and practice
EUROCRYPT'11 Proceedings of the 30th Annual international conference on Theory and applications of cryptographic techniques: advances in cryptology
Making NTRU as secure as worst-case problems over ideal lattices
EUROCRYPT'11 Proceedings of the 30th Annual international conference on Theory and applications of cryptographic techniques: advances in cryptology
Algorithms for the shortest and closest lattice vector problems
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
Analyzing blockwise lattice algorithms using dynamical systems
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
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
A parallel implementation of GaussSieve for the shortest vector problem in lattices
PaCT'11 Proceedings of the 11th international conference on Parallel computing technologies
Extreme enumeration on GPU and in clouds: how many dollars you need to break SVP challenges
CHES'11 Proceedings of the 13th international conference on Cryptographic hardware and embedded systems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Lattice enumeration using extreme pruning
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
BKZ 2.0: better lattice security estimates
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
The leakage-resilience limit of a computational problem is equal to its unpredictability entropy
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
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 O(1/ε2)n-time sieving algorithm for approximate integer programming
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Random walks and bisections in random circulant graphs
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Functional encryption for threshold functions (or fuzzy IBE) from lattices
PKC'12 Proceedings of the 15th international conference on Practice and Theory in Public Key Cryptography
Fully anonymous attribute tokens from lattices
SCN'12 Proceedings of the 8th international conference on Security and Cryptography for Networks
Attribute-based encryption for circuits
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Reusable garbled circuits and succinct functional encryption
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
On Ideal Lattices and Learning with Errors over Rings
Journal of the ACM (JACM)
Improvements in closest point search based on dual HKZ-bases
Theoretical Computer Science
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We give deterministic ~O(22n+o(n))-time algorithms to solve all the most important computational problems on point lattices in NP, including the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), and Shortest Independent Vectors Problem (SIVP). This improves the nO(n) running time of the best previously known algorithms for CVP (Kannan, Math. Operation Research 12(3):415--440, 1987) and SIVP (Micciancio, Proc. of SODA, 2008), and gives a deterministic and asymptotically faster alternative to the 2O(n)-time (and space) randomized algorithm for SVP of (Ajtai, Kumar and Sivakumar, STOC 2001). The core of our algorithm is a new method to solve the closest vector problem with preprocessing (CVPP) that uses the Voronoi cell of the lattice (described as intersection of half-spaces) as the result of the preprocessing function. In the process, we also give algorithms for several other lattice problems, including computing the kissing number of a lattice, and computing the set of all Voronoi relevant vectors. All our algorithms are deterministic, and have 2O(n) time and space complexity.