N-Dimensional LLL Reduction Algorithm with Pivoted Reflection

N-Dimensional LLL Reduction Algorithm with Pivoted Reflection

The Lenstra-Lenstra-Lovász (LLL) lattice reduction algorithm and many of its variants have been widely used by cryptography, multiple-input-multiple-output (MIMO) communication systems and carrier phase positioning in global navigation satellite system (GNSS) to solve the integer least squares (ILS) problem. In this paper, we propose an n-dimensional LLL reduction algorithm (n-LLL), expanding t...

An LLL-Reduction Algorithm with Quasi-linear Time Complexity1

We devise an algorithm, e L, with the following specifications: It takes as input an arbitrary basis B = (bi)i ∈ Zd×d of a Euclidean lattice L; It computes a basis of L which is reduced for a mild modification of the Lenstra-Lenstra-Lovász reduction; It terminates in time O(dβ + dβ) where β = log max ‖bi‖ (for any ε > 0 and ω is a valid exponent for matrix multiplication). This is the first LLL...

An LLL-reduction algorithm with quasi-linear time complexity

We devise an algorithm, e L, with the following specifications: It takes as input an arbitrary basis B = (bi)i ∈ Zd×d of a Euclidean lattice L; It computes a basis of L which is reduced for a mild modification of the Lenstra-Lenstra-Lovász reduction; It terminates in time O(dβ + dβ) where β = log max ‖bi‖ (for any ε > 0 and ω is a valid exponent for matrix multiplication). This is the first LLL...

Partial LLL Reduction

The Lenstra-Lenstra-Lovasz (LLL) reduction has wide applications in digital communications. It can greatly improve the speed of the sphere decoding (SD) algorithms for solving an integer least squares (ILS) problem and the performance of the Babai integer point, a suboptimal solution to the ILS problem. Recently Ling and Howgrave-Graham proposed the so-called effective LLL (ELLL) reduction. It ...

Segment LLL-Reduction with Floating Point Orthogonalization