Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
A more efficient algorithm for lattice basis reduction
Journal of Algorithms
Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems
FCT '91 Proceedings of the 8th International Symposium on Fundamentals of Computation Theory
An improved low-density subset sum algorithm
EUROCRYPT'91 Proceedings of the 10th annual international conference on Theory and application of cryptographic techniques
The Two Faces of Lattices in Cryptology
CaLC '01 Revised Papers from the International Conference on Cryptography and Lattices
Using the inhomogeneous simultaneous approximation problem for cryptographic design
AFRICACRYPT'11 Proceedings of the 4th international conference on Progress in cryptology in Africa
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
Parallel shortest lattice vector enumeration on graphics cards
AFRICACRYPT'10 Proceedings of the Third international conference on Cryptology in Africa
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Let N be an integer with at least two distinct prime factors. We reduce the problem of factoring N to the task of finding random integer solutions (e1,..., et) ∈ ZZt of the inequalities |Σi=1tei log pi - log N|≤ N-c and Σi=1t|ei log pi| ≤ (2c-1)log N+o(log pt), where c 1 is fixed and p1, ..., pt are the first t primes. We show, under the assumption that the smooth integers distribute "uniformly", that there are Ne+o(1) many solutions (e1,...,et) if c 1 and if Ɛ := c - 1 - (2c - 1) log log N / log pt 0. We associate with the primes p1,...,pt a lattice L ⊂ Rt+1 of dimension t and we associate with N a point N ∈ Rt+1. We reduce the problem of factoring N to the task of finding random lattice vectors z that are sufficiently close to N in both the ∞-norm and thr 1-norm. The dimension t of the lattice L is polynomial in log N. For N ≅ 2512 it is about 6300. We also reduce the problem of computing, for a prime N, discrete logarithms of the units in ZZ/NZZ to a similar diophantiue approximation problem.