, be given by the columns of then the reduced basis is which is size-reduced, satisfies the Lovász condition, and is hence LLL-reduced, as described above. See W. Bosma. for details of the reduction process. Example from Z[i]4 Likewise, for the basis over the complex integers given by the columns of the matrix below, then the columns of the matrix below give an LLL-reduced basis. Implementations LLL is implemented in Arageli as the function lll_reduction_int fpLLL as a stand-alone implementation FLINT as the function fmpz_lll GAP as the function LLLReducedBasis Macaulay2 as the function LLL in the package LLLBases Magma as the functions LLL and LLLGram (taking a gram matrix) Maple as the function IntegerRelations[LLL] Mathematica as the function LatticeReduce Number Theory Library (NTL) as the function LLL PARI/GP as the function qflll Pymatgen as the function analysis.get_lll_reduced_lattice SageMath as the method LLL driven by fpLLL and NTL Isabelle/HOL in the 'archive of formal proofs' entry LLL_Basis_Reduction. This code exports to efficiently executable Haskell. See also Coppersmith method Notes References Theory of cryptography Computational number theory Lattice points