A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
Generating hard instances of lattice problems (extended abstract)
STOC '96 Proceedings of the twenty-eighth 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
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
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
An Improved Worst-Case to Average-Case Connection for Lattice Problems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
New lattice based cryptographic constructions
Proceedings of the thirty-fifth annual ACM symposium on Theory of 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
Fully homomorphic encryption using ideal lattices
Proceedings of the forty-first annual ACM symposium on Theory of computing
Toward basing fully homomorphic encryption on worst-case hardness
CRYPTO'10 Proceedings of the 30th annual conference on Advances in cryptology
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
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(MATH) We define a new family of collision resistant hash functions whose security is based on the worst case hardness of approximating the covering radius of a lattice within a factor O(&pgr;n2log n), where &pgr; is a value between 1 and √ \over n that depends on the solution of the closest vector problem in certain "almost perfect" lattices. Even for &pgr; = √ \over n, this improves the smallest (worst-case) inapproximability factor for lattice problems known to imply the existence of one-way functions. (Previously known best factor was O(n3+&egr;) for the shortest independent vector problem, due to Cai and Nerurkar, based on work of Ajtai.) Using standard transference theorems from the geometry of numbers, our result immediately gives a connection between the worst-case and average-case complexity of the shortest vector problem with connection factor O(&pgr;n3}log n), improving the best previously known connection factor O(n4+&egr;), also due to Ajtai, Cai and Nerurkar.