Solving low-density subset sum problems
Journal of the ACM (JACM)
Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
Succinct certificates for almost all subset sum problems
SIAM Journal on Computing
Generating hard instances of lattice problems (extended abstract)
STOC '96 Proceedings of the twenty-eighth 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
Random Lattices and a Conjectured 0 - 1 Law about Their Polynomial Time Computable Properties
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Lattice Reduction by Random Sampling and Birthday Methods
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
ACM Transactions on Algorithms (TALG)
Fast LLL-type lattice reduction
Information and Computation
Finding short lattice vectors within mordell's inequality
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Fast LLL-type lattice reduction
Information and Computation
Improved analysis of Kannan's shortest lattice vector algorithm
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Algorithms for the shortest and closest lattice vector problems
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
Rankin's constant and blockwise lattice reduction
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
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Schnorr's algorithm for finding an approximation for the shortest nonzero vector in an n dimensional lattice depends on a parameter k. He proved that for a fixed k ≤ n his algorithm (block 2k-reduction) provides a lattice vector whose length is greater than the length of a shortest nonzero vector in the lattice by at most a factor of (4k2) n/k. (The time required by the algorithm depends on k.) We show that if k=o(n), this bound on the performance of Schnorr's algorithm cannot be improved (apart from a constant factor in the exponent), namely there is a lattice and a basis so that if they are given as an input to the algorithm then the resulting approximating factor of the output is at least k ε n/k. (For larger integers k if Schnorr's algorithm runs in polynomial time then we have already a polynomial time algorithm for finding the shortest nonzero vector.) We also solve an open problem formulated by Schnorr about the the Korkine-Zolotareff lattice constants αk. We show that his upper bound αk ≤ k1 + ln k is the best possible apart from a constant factor in the exponent. We prove a similar result about his upper bound βk≤ 4k2, where βk is another lattice constant with an important role in Schnorr's analysis of his algorithm.