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
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 the SVP to within a factor (1+1/dimE) is NP-Hard under randomized reductions
Journal of Computer and System Sciences
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
Generating Hard Instances of the Short Basis Problem
ICAL '99 Proceedings of the 26th International Colloquium on Automata, Languages and Programming
Improving Lattice Based Cryptosystems Using the Hermite Normal Form
CaLC '01 Revised Papers from the International Conference on Cryptography and Lattices
An Improved Worst-Case to Average-Case Connection for Lattice Problems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Almost Perfect Lattices, the Covering Radius Problem, and Applications to Ajtai's Connection Factor
SIAM Journal on Computing
Hardness of approximating the shortest vector problem in lattices
Journal of the ACM (JACM)
Hardness of approximating the Shortest Vector Problem in high ℓp norms
Journal of Computer and System Sciences - Special issue on FOCS 2003
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
Worst-Case to Average-Case Reductions Based on Gaussian Measures
SIAM Journal on Computing
Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions
Computational Complexity
Lossy trapdoor functions and their applications
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Trapdoors for hard lattices and new cryptographic constructions
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The complexity of the covering radius problem
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
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
A Note on the Distribution of the Distance from a Lattice
Discrete & Computational Geometry
Fully homomorphic encryption using ideal lattices
Proceedings of the forty-first annual ACM symposium on Theory of computing
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)
On Bounded Distance Decoding, Unique Shortest Vectors, and the Minimum Distance Problem
CRYPTO '09 Proceedings of the 29th Annual International Cryptology Conference on Advances in Cryptology
Fast Cryptographic Primitives and Circular-Secure Encryption Based on Hard Learning Problems
CRYPTO '09 Proceedings of the 29th Annual International Cryptology Conference on Advances in Cryptology
Zero-Knowledge Protocols for NTRU: Application to Identification and Proof of Plaintext Knowledge
ProvSec '09 Proceedings of the 3rd International Conference on Provable Security
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
Smooth Projective Hashing and Password-Based Authenticated Key Exchange from Lattices
ASIACRYPT '09 Proceedings of the 15th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Multi-bit cryptosystems based on lattice problems
PKC'07 Proceedings of the 10th international conference on Practice and theory in public-key cryptography
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
Asymptotically efficient lattice-based digital signatures
TCC'08 Proceedings of the 5th conference on Theory of cryptography
The Learning with Errors Problem (Invited Survey)
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Lattice basis delegation in fixed dimension and shorter-ciphertext hierarchical IBE
CRYPTO'10 Proceedings of the 30th annual conference on Advances in cryptology
Toward basing fully homomorphic encryption on worst-case hardness
CRYPTO'10 Proceedings of the 30th annual conference on Advances in cryptology
Additively homomorphic encryption with d-operand multiplications
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
Adaptively secure identity-based identification from lattices without random oracles
SCN'10 Proceedings of the 7th international conference on Security and cryptography for networks
An improvement of key generation algorithm for Gentry's homomorphic encryption scheme
IWSEC'10 Proceedings of the 5th international conference on Advances in information and computer security
Improved zero-knowledge identification with lattices
ProvSec'10 Proceedings of the 4th international conference on Provable security
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
Linearly homomorphic signatures over binary fields and new tools for lattice-based signatures
PKC'11 Proceedings of the 14th international conference on Practice and theory in public key cryptography conference on Public key cryptography
Generating Shorter Bases for Hard Random Lattices
Theory of Computing Systems
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
Homomorphic signatures for polynomial functions
EUROCRYPT'11 Proceedings of the 30th Annual international conference on Theory and applications of cryptographic techniques: advances in cryptology
Semi-homomorphic encryption and multiparty computation
EUROCRYPT'11 Proceedings of the 30th Annual international conference on Theory and applications of cryptographic techniques: advances in cryptology
Pseudorandom knapsacks and the sample complexity of LWE search-to-decision reductions
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
Fully homomorphic encryption from ring-LWE and security for key dependent messages
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
Bi-deniable public-key encryption
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
Generalized compact knapsacks are collision resistant
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part II
Threshold decryption and zero-knowledge proofs for lattice-based cryptosystems
TCC'10 Proceedings of the 7th international conference on Theory of Cryptography
Public-key cryptographic primitives provably as secure as subset sum
TCC'10 Proceedings of the 7th international conference on Theory of Cryptography
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
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
Fully homomorphic encryption over the integers
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
Bounded key-dependent message security
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
A simple BGN-Type cryptosystem from LWE
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
Efficient lattice (H)IBE in the standard model
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
PQCrypto'10 Proceedings of the Third international conference on Post-Quantum Cryptography
Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices
TCC'06 Proceedings of the Third conference on Theory of Cryptography
Efficient identity-based encryption without pairings and key escrow for mobile devices
WASA'13 Proceedings of the 8th international conference on Wireless Algorithms, Systems, and Applications
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Lattice cryptography is one of the hottest and fastest moving areas in mathematical cryptography today. Interest in lattice cryptography is due to several concurring factors. On the theoretical side, lattice cryptography is supported by strong worst-case/average-case security guarantees. On the practical side, lattice cryptography has been shown to be very versatile, leading to an unprecedented variety of applications, from simple (and efficient) hash functions, to complex and powerful public key cryptographic primitives, culminating with the celebrated recent development of fully homomorphic encryption. Still, one important feature of lattice cryptography is simplicity: most cryptographic operations can be implemented using basic arithmetic on small numbers, and many cryptographic constructions hide an intuitive and appealing geometric interpretation in terms of point lattices. So, unlike other areas of mathematical cryptology even a novice can acquire, with modest effort, a good understanding of not only the potential applications, but also the underlying mathematics of lattice cryptography. In these notes, we give an introduction to the mathematical theory of lattices, describe the main tools and techniques used in lattice cryptography, and present an overview of the wide range of cryptographic applications. This material should be accessible to anybody with a minimal background in linear algebra and some familiarity with the computational framework of modern cryptography, but no prior knowledge about point lattices.