Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions
Computational Complexity
Special Issue On Worst-case Versus Average-case Complexity Editors' Foreword
Computational Complexity
Efficient reductions among lattice problems
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
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
Noninteractive Statistical Zero-Knowledge Proofs for Lattice Problems
CRYPTO 2008 Proceedings of the 28th Annual conference on Cryptology: Advances in Cryptology
A Framework for Efficient and Composable Oblivious Transfer
CRYPTO 2008 Proceedings of the 28th Annual conference on Cryptology: Advances in Cryptology
Explicit Hard Instances of the Shortest Vector Problem
PQCrypto '08 Proceedings of the 2nd International Workshop on Post-Quantum Cryptography
Concurrently Secure Identification Schemes Based on the Worst-Case Hardness of Lattice Problems
ASIACRYPT '08 Proceedings of the 14th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Public-key cryptosystems from the worst-case shortest vector problem: extended abstract
Proceedings of the forty-first annual ACM symposium on Theory of computing
On lattices, learning with errors, random linear codes, and cryptography
Journal of the ACM (JACM)
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
Efficient Public Key Encryption Based on Ideal Lattices
ASIACRYPT '09 Proceedings of the 15th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
EUROCRYPT'08 Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology
Lattice-based identification schemes secure under active attacks
PKC'08 Proceedings of the Practice and theory in public key cryptography, 11th international conference on Public key cryptography
Proceedings of the forty-second ACM symposium on Theory of computing
Faster exponential time algorithms for the shortest vector problem
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
An efficient and parallel Gaussian sampler for lattices
CRYPTO'10 Proceedings of the 30th annual conference on Advances in cryptology
Lattice basis delegation in fixed dimension and shorter-ciphertext hierarchical IBE
CRYPTO'10 Proceedings of the 30th annual conference on Advances in cryptology
A lattice-based threshold ring signature scheme
LATINCRYPT'10 Proceedings of the First international conference on Progress in cryptology: cryptology and information security in Latin America
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
Improved zero-knowledge identification with lattices
ProvSec'10 Proceedings of the 4th international conference on Provable security
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
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
Can homomorphic encryption be practical?
Proceedings of the 3rd ACM workshop on Cloud computing security workshop
Ring signature schemes from lattice basis delegation
ICICS'11 Proceedings of the 13th international conference on Information and communications security
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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
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-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Lattice signatures without trapdoors
EUROCRYPT'12 Proceedings of the 31st Annual international conference on Theory and Applications of Cryptographic Techniques
Circular and KDM security for identity-based encryption
PKC'12 Proceedings of the 15th international conference on Practice and Theory in Public Key Cryptography
Strongly secure authenticated key exchange from factoring, codes, and lattices
PKC'12 Proceedings of the 15th international conference on Practice and Theory in Public Key Cryptography
Inner-product lossy trapdoor functions and applications
ACNS'12 Proceedings of the 10th international conference on Applied Cryptography and Network Security
ID-Based signatures from lattices in the random oracle model
WISM'12 Proceedings of the 2012 international conference on Web Information Systems and Mining
Zero-Knowledge proofs with low amortized communication from lattice assumptions
SCN'12 Proceedings of the 8th international conference on Security and Cryptography for Networks
Fully anonymous attribute tokens from lattices
SCN'12 Proceedings of the 8th international conference on Security and Cryptography for Networks
A ciphertext policy attribute-based encryption scheme without pairings
Inscrypt'11 Proceedings of the 7th international conference on Information Security and Cryptology
On the number of lattice points in a small sphere and a recursive lattice decoding algorithm
Designs, Codes and Cryptography
Classical hardness of learning with errors
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Efficient hierarchical identity-based signatures from lattices
International Journal of Electronic Security and Digital Forensics
Secure pattern matching using somewhat homomorphic encryption
Proceedings of the 2013 ACM workshop on Cloud computing security workshop
How to share a lattice trapdoor: threshold protocols for signatures and (H)IBE
ACNS'13 Proceedings of the 11th international conference on Applied Cryptography and Network Security
On Ideal Lattices and Learning with Errors over Rings
Journal of the ACM (JACM)
Lattice-based message recovery signature schemes
International Journal of Electronic Security and Digital Forensics
On the probability of generating a lattice
Journal of Symbolic Computation
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We show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension of the lattice. The lattice problems we consider are the shortest vector problem, the shortest independent vectors problem, the covering radius problem, and the guaranteed distance decoding problem (a variant of the well-known closest vector problem). The approximation factor we obtain is $n \log^{O(1)} n$ for all four problems. This greatly improves on all previous work on the subject starting from Ajtai’s seminal paper [Generating hard instances of lattice problems, in Complexity of Computations and Proofs, Quad. Mat. 13, Dept. Math., Seconda Univ. Napoli, Caserta, Italy, 2004, pp. 1-32] up to the strongest previously known results by Micciancio [SIAM J. Comput., 34 (2004), pp. 118-169]. Our results also bring us closer to the limit where the problems are no longer known to be in NP intersect coNP. Our main tools are Gaussian measures on lattices and the high-dimensional Fourier transform. We start by defining a new lattice parameter which determines the amount of Gaussian noise that one has to add to a lattice in order to get close to a uniform distribution. In addition to yielding quantitatively much stronger results, the use of this parameter allows us to simplify many of the complications in previous work. Our technical contributions are twofold. First, we show tight connections between this new parameter and existing lattice parameters. One such important connection is between this parameter and the length of the shortest set of linearly independent vectors. Second, we prove that the distribution that one obtains after adding Gaussian noise to the lattice has the following interesting property: the distribution of the noise vector when conditioning on the final value behaves in many respects like the original Gaussian noise vector. In particular, its moments remain essentially unchanged.