Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
Complexity of Lattice Problems
Complexity of Lattice Problems
Closest Vectors, Successive Minima, and Dual HKZ-Bases of Lattices
ICALP '00 Proceedings of the 27th International Colloquium on Automata, Languages and Programming
The LLL Algorithm: Survey and Applications
The LLL Algorithm: Survey and Applications
Improved analysis of Kannan's shortest lattice vector algorithm
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Proceedings of the forty-second ACM symposium on Theory of computing
Closest point search in lattices
IEEE Transactions on Information Theory
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In this paper we review the technique to solve the CVP based on dual HKZ-bases by J. Blomer [4]. The technique is based on the transference theorems given by Banaszczyk [3] which imply some necessary conditions on the coefficients of the closest vectors with respect to a basis whose dual is HKZ-reduced. Recursively, starting with the last coefficient, intervals of length i can be derived for the i-th coefficient of any closest vector. This leads to n! candidates for closest vectors. In this paper we refine the necessary conditions derived from the transference theorems, giving an exponential reduction of the number of candidates. The improvement is due to the fact that the lengths of the intervals are not independent. In the original algorithm the candidates for a coefficient pair (v"i,v"i"+"1) correspond to the integer points in a rectangle of volume i@?(i+1). In our analysis we show that the candidates for (v"i,v"i"+"1) in fact lie in an ellipse with transverse and conjugate diameter i+1, respectively i. This reduces the expected number of points to be enumerated by an exponential factor of about 0.886^n. We further show how a choice of the coefficients (v"n,...,v"i"+"1) influences the interval from which v"i can be chosen. Numerical computations show that these considerations allow to bound the expected number of points to be enumerated by n^0^.^7^5^n for 10=