Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
A course in computational algebraic number theory
A course in computational algebraic number theory
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
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
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
Discrete Applied Mathematics - Special issue: Special issue devoted to the fifth annual international computing and combinatories conference (COCOON'99) Tokyo, Japan 26-28 July 1999
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
A Table of Totally Complex Number Fields of Small Discriminants
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
An Improved Worst-Case to Average-Case Connection for Lattice Problems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
The inapproximability of lattice and coding problems with preprocessing
Journal of Computer and System Sciences - Special issue on computational complexity 2002
Worst-Case to Average-Case Reductions Based on Gaussian Measures
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Almost Perfect Lattices, the Covering Radius Problem, and Applications to Ajtai's Connection Factor
SIAM Journal on Computing
New lattice-based cryptographic constructions
Journal of the ACM (JACM)
On lattices, learning with errors, random linear codes, and cryptography
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Fast quantum algorithms for computing the unit group and class group of a number field
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Journal of the ACM (JACM)
Hardness of approximating the shortest vector problem in lattices
Journal of the ACM (JACM)
Lattice problems and norm embeddings
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
On basing one-way functions on NP-hardness
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
Limits on the Hardness of Lattice Problems in \ell _p Norms
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Generalized compact knapsacks are collision resistant
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part II
Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices
TCC'06 Proceedings of the Third conference on Theory of Cryptography
The hardness of the closest vector problem with preprocessing
IEEE Transactions on Information Theory
Constructions of codes from number fields
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
SWIFFT: A Modest Proposal for FFT Hashing
Fast Software Encryption
Fully homomorphic encryption using ideal lattices
Proceedings of the forty-first annual ACM symposium on Theory of computing
Fiat-Shamir with Aborts: Applications to Lattice and Factoring-Based Signatures
ASIACRYPT '09 Proceedings of the 15th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Asymptotically efficient lattice-based digital signatures
TCC'08 Proceedings of the 5th conference on Theory of cryptography
Algorithms for ray class groups and Hilbert class fields
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Toward basing fully homomorphic encryption on worst-case hardness
CRYPTO'10 Proceedings of the 30th annual conference on Advances in cryptology
Implementing Gentry's fully-homomorphic encryption scheme
EUROCRYPT'11 Proceedings of the 30th Annual international conference on Theory and applications of cryptographic techniques: advances in cryptology
The geometry of lattice cryptography
Foundations of security analysis and design VI
Fully homomorphic encryption from ring-LWE and security for key dependent messages
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
On ideal lattices and learning with errors over rings
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
Bonsai trees, or how to delegate a lattice basis
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
On Ideal Lattices and Learning with Errors over Rings
Journal of the ACM (JACM)
Hi-index | 0.00 |
We exhibit an average-case problem that is as hard as finding γ(n)-approximate shortest nonzero vectors in certain n-dimensional lattices in the worst case, for γ(n) = O(√log n). The previously best known factor for any non-trivial class of lattices was γ(n) = Õ(n). Our results apply to families of lattices having special algebraic structure. Specifically, we consider lattices that correspond to ideals in the ring of integers of an algebraic number field. The worst-case problem we rely on is to find approximate shortest vectors in these lattices, under an appropriate form of preprocessing of the number field. For the connection factors γ(n) we achieve, the corresponding decision problems on ideal lattices are not known to be NP-hard; in fact, they are in P. However, the search approximation problems still appear to be very hard. Indeed, ideal lattices are well-studied objects in computational number theory, and the best known algorithms for them seem to perform no better than the best known algorithms for general lattices. To obtain the best possible connection factor, we instantiate our constructions with infinite families of number fields having constant root discriminant. Such families are known to exist and are computable, though no efficient construction is yet known. Our work motivates the search for such constructions. Even constructions of number fields having root discriminant up to O(n2/3-ε) would yield connection factors better than Õ(n). As an additional contribution, we give reductions between various worst-case problems on ideal lattices, showing for example that the shortest vector problem is no harder than the closest vector problem. These results are analogous to previously-known reductions for general lattices.