In this paper we construct multi-dimensional p-adic approximation lattices by simultaneous rational approximations of p-adic numbers. For analyzing these p-adic lattices we apply the LLL algorithm due to Lenstra, Lenstra and Lovász, which has been widely used to solve the various NP problems such as SVP (Shortest Vector Problems), ILP (Integer Linear Programing) .. and so on. In a twodimensiona...

The LLL algorithm aims at finding a “reduced” basis of a Euclidean lattice. The LLL algorithm plays a primary role in many areas of mathematics and computer science. However, its general behaviour is far from being well understood. There are already many experimental observations about the number of iterations or the geometry of the output, that pose challenging questions that remain unanswered...

Multiple-input multiple-output (MIMO) systems are playing an important role in the recent wireless communication. The complexity of the different systems models challenge different researches to get a good complexity to performance balance. Lattices Reduction Techniques and Lenstra-Lenstra-Lovàsz (LLL) algorithm bring more resources to investigate and can contribute to the complexity reduction ...

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Let B be a basis of a Euclidean lattice, and B̃ an approximation thereof. We give a sufficient condition on the closeness between B̃ and B so that an LLL-reducing transformation U for B̃ remains valid for B. Further, we analyse an efficient reduction algorithm when B is itself a small deformation of an LLL-reduced basis. Applications include speeding-up reduction by keeping only the most significa...

The Lenstra–Lenstra–Lovász lattice basis reduction algorithm (called LLL or L3) is a fundamental tool in computational number theory and theoretical computer science, which can be viewed as an efficient algorithmic version of Hermite’s inequality on Hermite’s constant. Given an integer d-dimensional lattice basis with vectors of Euclidean norm less than B in an ndimensional space, the L3 algori...

In this paper, three practical lattice basis reduction algorithms are presented. The first algorithm constructs a Hermite, Korkine and Zolotareff (HKZ) reduced lattice basis, in which a unimodular transformation is used for basis expansion. Our complexity analysis shows that our algorithm is significantly more efficient than the existing HKZ reduction algorithms. The second algorithm computes a...