The Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and László Lovász in 1982. Given a basis with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with , the LLL algorithm calculates an LLL-reduced (short, nearly orthogonal) lattice basis in time where is the largest length of under the Euclidean norm, that is, . The original applications were to give polynomial-time algorithms for factorizing polynomials with rational coefficients, for finding simultaneous rational approximations to real numbers, and for solving the integer linear programming problem in fixed dimensions. LLL reduction The precise definition of LLL-reduced is as follows: Given a basis define its Gram–Schmidt process orthogonal basis and the Gram-Schmidt coefficients for any . Then the basis is LLL-reduced if there exists a parameter in such that the following holds: (size-reduced) For . By definition, this property guarantees the length reduction of the ordered basis. (Lovász condition) For k = 2,3,..,n . Here, estimating the value of the parameter, we can conclude how well the basis is reduced. Greater values of lead to stronger reductions of the basis. Initially, A. Lenstra, H. Lenstra and L. Lovász demonstrated the LLL-reduction algorithm for . Note that although LLL-reduction is well-defined for , the polynomial-time complexity is guaranteed only for in . The LLL algorithm computes LLL-reduced bases. There is no known efficient algorithm to compute a basis in which the basis vectors are as short as possible for lattices of dimensions greater than 4. However, an LLL-reduced basis is nearly as short as possible, in the sense that there are absolute bounds such that the first basis vector is no more than times as long as a shortest vector in the lattice, the second basis vector is likewise within of the second successive minimum, and so on. Applications An early successful application of the LLL algorithm was its use by Andrew Odlyzko and Herman te Riele in disproving Mertens conjecture. The LLL algorithm has found numerous other applications in MIMO detection algorithms and cryptanalysis of public-key encryption schemes: knapsack cryptosystems, RSA with particular settings, NTRUEncrypt, and so forth. The algorithm can be used to find integer solutions to many problems. In particular, the LLL algorithm forms a core of one of the integer relation algorithms. For example, if it is believed that r=1.618034 is a (slightly rounded) root