This paper presents improved practical algorithms for lattice basis reduction, focusing on the $L^3$-algorithm and its variants. The authors propose a practical floating-point version of the $L^3$-algorithm, which is more stable and efficient than the original algorithm. They also introduce a variant of the $L^3$-algorithm with "deep insertions" and a practical algorithm for block Korkin–Zolotarev reduction. These algorithms are shown to solve almost all subset sum problems with up to 66 random weights of arbitrary bit length within a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC 1+ computer. The paper discusses the theoretical foundations of lattice basis reduction, including the $L^3$-algorithm and its variants, and provides empirical tests to demonstrate the effectiveness of the proposed algorithms. The algorithms are applied to solve the knapsack or subset sum problem, and the authors compare their performance with existing methods, showing significant improvements in success rates.This paper presents improved practical algorithms for lattice basis reduction, focusing on the $L^3$-algorithm and its variants. The authors propose a practical floating-point version of the $L^3$-algorithm, which is more stable and efficient than the original algorithm. They also introduce a variant of the $L^3$-algorithm with "deep insertions" and a practical algorithm for block Korkin–Zolotarev reduction. These algorithms are shown to solve almost all subset sum problems with up to 66 random weights of arbitrary bit length within a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC 1+ computer. The paper discusses the theoretical foundations of lattice basis reduction, including the $L^3$-algorithm and its variants, and provides empirical tests to demonstrate the effectiveness of the proposed algorithms. The algorithms are applied to solve the knapsack or subset sum problem, and the authors compare their performance with existing methods, showing significant improvements in success rates.