Improved cryptographic hash functions with worst-case/average-case connection
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Some facets of complexity theory and cryptography: A five-lecture tutorial
ACM Computing Surveys (CSUR)
The inapproximability of lattice and coding problems with preprocessing
Journal of Computer and System Sciences - Special issue on computational complexity 2002
ACM Transactions on Algorithms (TALG)
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
Fast LLL-type lattice reduction
Information and Computation
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
Foundations and Trends® in Theoretical Computer Science
Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions
Computational Complexity
Efficient reductions among lattice problems
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Limits on the Hardness of Lattice Problems in lp Norms
Computational Complexity
Sampling methods for shortest vectors, closest vectors and successive minima
Theoretical Computer Science
New NP-Complete Problems Associated with Lattices
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
A Digital Signature Scheme Based on NP-Complete Lattice Problems
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
A deterministic reduction for the gap minimum distance problem: [extended abstract]
Proceedings of the forty-first annual ACM symposium on Theory of computing
Fast LLL-type lattice reduction
Information and Computation
Proceedings of the forty-second ACM symposium on Theory of computing
NTRU-like public key cryptosystems beyond dedekind domain up to alternative algebra
Transactions on computational science X
Practical polynomial factoring in polynomial time
Proceedings of the 36th international symposium on Symbolic and algebraic computation
A simple deterministic reduction for the gap minimum distance of code problem
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
The geometry of lattice cryptography
Foundations of security analysis and design VI
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
Gradual sub-lattice reduction and a new complexity for factoring polynomials
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
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
Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices
TCC'06 Proceedings of the Third conference on Theory of Cryptography
Lower bounds of shortest vector lengths in random NTRU lattices
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
Sampling methods for shortest vectors, closest vectors and successive minima
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
| Hi-index | 0.00 |
We show that approximating the shortest vector problem (in any $\ell_p$ norm) to within any constant factor less than $\sqrt[p]2$ is hard for NP under reverse unfaithful random reductions with inverse polynomial error probability. In particular, approximating the shortest vector problem is not in RP (random polynomial time), unless NP equals RP. We also prove a proper NP-hardness result (i.e., hardness under deterministic many-one reductions) under a reasonable number theoretic conjecture on the distribution of square-free smooth numbers. As part of our proof, we give an alternative construction of Ajtai's constructive variant of Sauer's lemma that greatly simplifies Ajtai's original proof.