Solving low-density subset sum problems
Journal of the ACM (JACM)
Algorithms to construct Minkowski reduced and Hermite reduced lattice bases
Theoretical Computer Science
On the Lagarias-Odlyzko algorithm for the subset sum problem
SIAM Journal on Computing
Proceedings of CRYPTO 84 on Advances in cryptology
Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
Reconstructing truncated integer variables satisfying linear congruences
SIAM Journal on Computing - Special issue on cryptography
Solving simultaneous modular equations of low degree
SIAM Journal on Computing - Special issue on cryptography
A more efficient algorithm for lattice basis reduction
Journal of Algorithms
Succinct certificates for almost all subset sum problems
SIAM Journal on Computing
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
Handbook of combinatorics (vol. 1)
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
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
On the limits of non-approximability of lattice problems
STOC '98 Proceedings of the thirtieth 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
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
A lattice-based public-key cryptosystem
Information and Computation
Finding smooth integers in short intervals using CRT decoding
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Finding the closest lattice vector when it's unusually close
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Sparse polynomial approximation in finite fields
STOC '01 Proceedings of the thirty-third 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
Lattice Attacks on Digital Signature Schemes
Designs, Codes and Cryptography
Handbook of Applied Cryptography
Handbook of Applied Cryptography
How to Guess l-th Roots Modulo n by Reducing Lattice Bases
AAECC-6 Proceedings of the 6th International Conference, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
On the Generalised Hidden Number Problem and Bit Security of XTR
AAECC-14 Proceedings of the 14th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Cryptanalysis of a Fast Public Key Cryptosystem Presented at SAC '97
SAC '98 Proceedings of the Selected Areas in Cryptography
CRYPTO '99 Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology
The Hardness of the Hidden Subset Sum Problem and Its Cryptographic Implications
CRYPTO '99 Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology
Cryptanalysis of the Goldreich-Goldwasser-Halevi Cryptosystem from Crypto '97
CRYPTO '99 Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology
A Chosen-Ciphertext Attack against NTRU
CRYPTO '00 Proceedings of the 20th Annual International Cryptology Conference on Advances in Cryptology
On the Unpredictability of Bits of the Elliptic Curve Diffie--Hellman Scheme
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
RSA-OAEP Is Secure under the RSA Assumption
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
Hardness of Computing the Most Significant Bits of Secret Keys in Diffie-Hellman and Related Schemes
CRYPTO '96 Proceedings of the 16th Annual International Cryptology Conference on Advances in Cryptology
On the Security of the KMOV Public Key Cryptosystem
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
"Pseudo-Random" Number Generation Within Cryptographic Algorithms: The DDS Case
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
Public-Key Cryptosystems from Lattice Reduction Problems
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
Eliminating Decryption Errors in the Ajtai-Dwork Cryptosystem
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
A Multiplicative Attack Using LLL Algorithm on RSA Signatures with Redundancy
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
Cryptanalysis of the Chor-Rivest Cryptosystem
CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
Cryptanalysis of the Ajtai-Dwork Cryptosystem
CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
Key Recovery and Message Attacks on NTRU-Composite
EUROCRYPT '01 Proceedings of the International Conference on the Theory and Application of Cryptographic Techniques: Advances in Cryptology
An Attack on RSA Given a Small Fraction of the Private Key Bits
ASIACRYPT '98 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
The Béguin-Quisquater Server-Aided RSA Protocol from Crypto '95 is not Secure
ASIACRYPT '98 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
Why Textbook ElGamal and RSA Encryption Are Insecure
ASIACRYPT '00 Proceedings of the 6th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Cryptanalysis of the RSA Schemes with Short Secret Exponent from Asiacrypt '99
ASIACRYPT '00 Proceedings of the 6th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Finding Small Roots of Univariate Modular Equations Revisited
Proceedings of the 6th IMA International Conference on Cryptography and Coding
Tensor-Based Trapdoors for CVP and Their Application to Public Key Cryptography
Proceedings of the 7th IMA International Conference on Cryptography and Coding
The Effectiveness of Lattice Attacks Against Low-Exponent RSA
PKC '99 Proceedings of the Second International Workshop on Practice and Theory in Public Key Cryptography
Certificates of Recoverability with Scale Recovery Agent Security
PKC '00 Proceedings of the Third International Workshop on Practice and Theory in Public Key Cryptography: Public Key Cryptography
The Decision Diffie-Hellman Problem
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
NTRU: A Ring-Based Public Key Cryptosystem
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
A Montgomery-Like Square Root for the Number Field Sieve
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
The Complexity of Some Lattice Problems
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
Lattice Reduction in Cryptology: An Update
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
Improving Lattice Based Cryptosystems Using the Hermite Normal Form
CaLC '01 Revised Papers from the International Conference on Cryptography and Lattices
Approximate Integer Common Divisors
CaLC '01 Revised Papers from the International Conference on Cryptography and Lattices
Finding Small Solutions to Small Degree Polynomials
CaLC '01 Revised Papers from the International Conference on Cryptography and Lattices
The Insecurity of Nyberg-Rueppel and Other DSA-Like Signature Schemes with Partially Known Nonces
CaLC '01 Revised Papers from the International Conference on Cryptography and Lattices
Simplified OAEP for the RSA and Rabin Functions
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
An Improved Worst-Case to Average-Case Connection for Lattice Problems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Approximating-CVP to within Almost-Polynomial Factors is NP-Hard
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant
FOCS '98 Proceedings of the 39th 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
On breaking generalized knapsack public key cryptosystems
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
On the hardness of the shortest vector problem
On the hardness of the shortest vector problem
A polynomial time algorithm for breaking the basic Merkle-Hellman cryptosystem
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
EUROCRYPT'97 Proceedings of the 16th annual international conference on Theory and application of cryptographic techniques
Factoring integers and computing discrete logarithms via diophantine approximation
EUROCRYPT'91 Proceedings of the 10th annual international conference on Theory and application of cryptographic techniques
Attacking the Chor-Rivest cryptosystem by improved lattice reduction
EUROCRYPT'95 Proceedings of the 14th annual international conference on Theory and application of cryptographic techniques
Cryptanalysis of RSA with private key d less than N0:292
EUROCRYPT'99 Proceedings of the 17th international conference on Theory and application of cryptographic techniques
Factorization of a 512-bit RSA modulus
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
Noisy polynomial interpolation and noisy chinese remaindering
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
Cryptanalysis of countermeasures proposed for repairing ISO 9796-1
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
The hardness of the closest vector problem with preprocessing
IEEE Transactions on Information Theory
On the Generalised Hidden Number Problem and Bit Security of XTR
AAECC-14 Proceedings of the 14th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Hidden Number Problem with the Trace and Bit Security of XTR and LUC
CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
Analysis and Improvements of NTRU Encryption Paddings
CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
The Hardness of Hensel Lifting: The Case of RSA and Discrete Logarithm
ASIACRYPT '02 Proceedings of the 8th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Chinese Remaindering for Algebraic Numbers in a Hidden Field
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
Cryptanalysis of General Lu-Lee Type Systems
Information Security and Cryptology
Algebraic Cryptanalysis of CTRU Cryptosystem
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Solving Linear Equations Modulo Divisors: On Factoring Given Any Bits
ASIACRYPT '08 Proceedings of the 14th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Broadcast Attacks against Lattice-Based Cryptosystems
ACNS '09 Proceedings of the 7th International Conference on Applied Cryptography and Network Security
Attacking Power Generators Using Unravelled Linearization: When Do We Output Too Much?
ASIACRYPT '09 Proceedings of the 15th International Conference on the Theory and Application of Cryptology and Information Security: 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
Accelerating lattice reduction with FPGAs
LATINCRYPT'10 Proceedings of the First international conference on Progress in cryptology: cryptology and information security in Latin America
Lattice Polly Cracker cryptosystems
Journal of Symbolic Computation
Improved Nguyen-Vidick heuristic sieve algorithm for shortest vector problem
Proceedings of the 6th ACM Symposium on Information, Computer and Communications Security
Analyzing blockwise lattice algorithms using dynamical systems
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
Fully homomorphic encryption over the integers with shorter public keys
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
Modulus fault attacks against RSA-CRT signatures
CHES'11 Proceedings of the 13th international conference on Cryptographic hardware and embedded systems
On the provable security of an efficient RSA-Based pseudorandom generator
ASIACRYPT'06 Proceedings of the 12th international conference on Theory and Application of Cryptology and Information Security
Predicting subset sum pseudorandom generators
SAC'04 Proceedings of the 11th international conference on Selected Areas in Cryptography
Another look at “provable security”. II
INDOCRYPT'06 Proceedings of the 7th international conference on Cryptology in India
Spectral norm in learning theory: some selected topics
ALT'06 Proceedings of the 17th international conference on Algorithmic Learning Theory
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Adapting density attacks to low-weight knapsacks
ASIACRYPT'05 Proceedings of the 11th international conference on Theory and Application of Cryptology and Information Security
Cryptanalysis of the quadratic generator
INDOCRYPT'05 Proceedings of the 6th international conference on Cryptology in India
MaTRU: a new NTRU-Based cryptosystem
INDOCRYPT'05 Proceedings of the 6th international conference on Cryptology in India
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
Experimenting with faults, lattices and the DSA
PKC'05 Proceedings of the 8th international conference on Theory and Practice in Public Key Cryptography
EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
Fully homomorphic encryption with relatively small key and ciphertext sizes
PKC'10 Proceedings of the 13th international conference on Practice and Theory in Public Key Cryptography
Fully homomorphic encryption over the integers
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
A new key exchange protocol based on MQV assuming public computations
SCN'06 Proceedings of the 5th international conference on Security and Cryptography for Networks
Rankin's constant and blockwise lattice reduction
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
Symplectic lattice reduction and NTRU
EUROCRYPT'06 Proceedings of the 24th annual international conference on The Theory and Applications of Cryptographic Techniques
Learning a parallelepiped: cryptanalysis of GGH and NTRU signatures
EUROCRYPT'06 Proceedings of the 24th annual international conference on The Theory and Applications of Cryptographic Techniques
Practical modifications of leadbitter et al.'s repeated-bits side-channel analysis on (EC)DSA
WISA'05 Proceedings of the 6th international conference on Information Security Applications
Learning a zonotope and more: cryptanalysis of NTRUSign countermeasures
ASIACRYPT'12 Proceedings of the 18th international conference on The Theory and Application of Cryptology and Information Security
A Comparative Survey on Cryptology-Based Methodologies
International Journal of Information Security and Privacy
Sparse Non-negative Stencils for Anisotropic Diffusion
Journal of Mathematical Imaging and Vision
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Lattices are regular arrangements of points in n-dimensional space, whose study appeared in the 19th century in both number theory and crystallography. Since the appearance of the celebrated Lenstra-Lenstra-Lovász lattice basis reduction algorithm twenty years ago, lattices have had surprising applications in cryptology. Until recently, the applications of lattices to cryptology were only negative, as lattices were used to break various cryptographic schemes. Paradoxically, several positive cryptographic applications of lattices have emerged in the past five years: there now exist public-key cryptosystems based on the hardness of lattice problems, and lattices play a crucial rôle in a few security proofs. We survey the main examples of the two faces of lattices in cryptology.