The LLL algorithm using fast givens

In this paper, by applying the fast Givens transformation, we propose a square-root free LLL algorithm. Thus, like the original LLL algorithm, our algorithm can perform exact arithmetic when the basis vectors are integer. Moreover, our algorithm has the following advantages over the original and floating-point LLL algorithms: 1. Our algorithm is consistent in that it uses the fast Givens in both the upper triangularization stage and the basis reduction stage. 2. For real or complex lattice, the fast Givens QR decomposition is much more numerically stable than the Gram-Schmidt method used in the original LLL algorithm, while in the basis reduction stage, the fast Givens transformation is also much more stable than the nonsingular transformation used in the original LLL algorithm. 3. Since two entries of the 2-by-2 fast Givens matrix are both equal to 1, then the computational cost of our algorithm is much less than that of the floating-point LLL algorithm. Specifically, the number of multiplications required by our algorithm is only half of that of the floating-point LLL algorithm.