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 d \leq n, the LLL algorithm calculates an LLL-reduced (short, nearly orthogonal) lattice basis in time where B 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 B is LLL-reduced if there exists a parameter \delta in such that the following holds: Here, estimating the value of the \delta parameter, we can conclude how well the basis is reduced. Greater values of \delta 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 \delta = 1, the polynomial-time complexity is guaranteed only for \delta in (0.25,1). 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 c_i > 1 such that the first basis vector is no more than c_1 times as long as a shortest vector in the lattice, the second basis vector is likewise within c_2 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 to an unknown quadratic equation with integer coefficients, one may apply LLL reduction to the lattice in spanned by and. The first vector in the reduced basis will be an integer linear combination of these three, thus necessarily of the form ; but such a vector is "short" only if a, b, c are small and ar^2+br+c is even smaller. Thus the first three entries of this short vector are likely to be the coefficients of the integral quadratic polynomial which has r as a root. In this example the LLL algorithm finds the shortest vector to be [1, -1, -1, 0.00025] and indeed x^2-x-1 has a root equal to the golden ratio, 1.6180339887....

Properties of LLL-reduced basis

Let be a \delta-LLL-reduced basis of a lattice \mathcal L. From the definition of LLL-reduced basis, we can derive several other useful properties about \mathbf{B}.

LLL algorithm pseudocode

The following description is based on, with the corrections from the errata. INPUT a lattice basis b1, b2, ..., bn in Zm a parameter δ with 1/4 < δ < 1, most commonly δ = 3/4 PROCEDURE B* <- GramSchmidt({b1, ..., bn}) = {b1*, ..., bn*}; and do not normalize μi,j <- InnerProduct(bi, bj*)/InnerProduct(bj*, bj*); using the most current values of bi and bj* k <- 2; while k <= n do for j from k−1 to 1 do if |μk,j| > 1/2 then bk <- bk − ⌊μk,j⌉bj; Update B* and the related μi,j 's as needed. (The naive method is to recompute B* whenever ***b*****i changes: B <- GramSchmidt({b1, ..., bn}) = {b1, ..., bn}) end if end for if InnerProduct(bk, bk) > (δ − μ2k,k−1) InnerProduct(bk−1, bk−1) then k <- k + 1; else Swap bk and bk−1; Update B and the related μi,j 's as needed. k <- max(k−1, 2); end if end while return B the LLL reduced basis of {b1, ..., bn} OUTPUT the reduced basis b1, b2, ..., bn in Zm

Examples

Example from Z3

Let a lattice basis, 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