A Hybrid Method for Lattice Basis Reduction

Lattice reduction has a wide range of applications. In this paper, we first present a polynomial time Jacobi method for lattice basis reduction by modifying the condition for the Lagrange reduction and integrating the size reduction into the algorithm. We show that the complexity of the modified Jacobi algorithm is O(n5 logB), where n is the dimension of the lattice and B is the maximum length of the input lattice basis vectors. To improve the quality of the computed bases, we then enhance our method by including a postprocessing without compromising the complexity. Our experiments show that our hybrid algorithm computes better reduced bases than the well-known LLL algorithm in terms of both the orthogonality defect and the condition number of the basis matrix. Moreover, although our algorithm has higher complexity than the LLL algorithm, it runs faster for problems of sizes under 90.