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...

The Lenstra-Lenstra-Lovász (LLL) algorithm is the most practical lattice reduction algorithm in digital communications. In this paper, several variants of the LLL algorithm with either lower theoretic complexity or fixed-complexity implementation are proposed and/or analyzed. Firstly, the O(n log n) theoretic average complexity of the standard LLL algorithm under the model of i.i.d. complex nor...

In this paper, we present a delayed size-reduction technique for speeding up the LLL algorithm. It can significantly speed up the LLL algorithm without sacrificing the quality of the results. Our experiments have shown that for problems of size 80, our algorithm can be twice as fast as the LLL algorithm. For larger size problems, the speed up is greater. Moreover, our algorithm provides a start...

The LLL algorithm is widely used to solve the integer least squares problems that arise in many engineering applications. As most practitioners did not understand how the LLL algorithm works, they avoided the issue by referring to the method as an integer Gram Schmidt approach (without explaining what they mean by this term). Luk and Tracy were first to describe the behavior of the LLL algorith...

Malocclusions, such as an open bite and high canines, are often encountered in orthodontic practice. Teeth without occlusal stimuli are known as hypofunctional teeth, and numerous atrophic changes have been reported in the periodontal tissue, including reductions in blood vessels in the periodontal ligament (PDL), heavy root resorption, and reduced bone mineral density (BMD) in the alveolar bon...

PURPOSE To characterize the effects of low-level laser (LLL) on third-degree burn wounds which were infected with Staphylococcus aureus (S. aureus) in diabetic rats. METHODS Thirty streptozotocin-induced diabetic rats were divided into two groups: the control and the LLL groups. Third-degree burns were induced using a heated metal rod, and then, were contaminated with S. aureus. The wounds in...

INTRODUCTION There have been inconsistent reports of dilation of the third ventricle (lll-V) in bipolar disorder. Within the lateral walls of lll-V are hypothalamic nuclei which mediate the neuroendocrine, sleep, appetite and autonomic disturbances which characterise a depressive episode. METHODS The lll-V width, immediately anterior to the mamillary bodies, was measured in 74 bipolar I or II...

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 ...