The Lenstra–Lenstra–Lovász lattice basis reduction (LLL) is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and László Lovász in 1982.^[1] Given as input d lattice basis vectors with n-dimensional integer coordinates and norms lesser than B, the LLL algorithm outputs an LLL-reduced (short, nearly orthogonal) lattice basis in time

The original applications were to give polynomial time algorithms for factorizing polynomials with rational coefficients into irreducible polynomials, for finding simultaneous rational approximations to real numbers, and for solving the integer linear programming problem in fixed dimensions.

Contents 1 LLL reduction 2 Applications 3 Implementations 4 Notes 5 References

[edit] LLL reduction

The precise definition of LLL-reduced is as follows: the basis

with its Gram–Schmidt process orthogonal basis,

is LLL-reduced if there exists a parameter δ in (0.25,1] such that

1. (size-reduced)
2. (Lovász condition)

where

, for any i>j.

Note that although LLL-reduction is well-defined for δ = 1, the polynomial-time complexity is guaranteed only for δ in (0.25,1).

The LLL algorithm computes LLL-reduced bases. A reduced basis is one in which the basis vectors are as short as possible. There is no known efficient algorithm to compute such a basis except for lattices of dimensions up to 4. A basis which is nearly reduced, in the sense that there are absolute bounds c_i > 1 such that the first basis vector is no more than c₁ times as long as a shortest vector in the lattice, the second basis vector is likewise within c₂ of the second successive minimum, and so on, is computed by the LLL algorithm.

[edit] Applications

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.^[2]

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 a quadratic equation with integer coefficients, one may apply the LLL reduction to the lattice in R⁴ spanned by [1,0,0,10000r²],[0,1,0,10000r], and [0,0,1,10000]. The first vector in the reduced basis will be an integer linear combination of these three, thus necessarily of the form [a,b,c,10000(ar² + br + c)]; but such a vector is "short" only if a, b, c are small and ar² + 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 x² − x − 1 has a root equal to 1.6180339887…

[edit] Implementations

LLL is implemented in

• Arageli as the function lll_reduction_int
• fpLLL as a stand-alone implementation
• GAP as the function LLLReducedBasis
• LiDIA as the function/method lll in the LT package
• 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
• SAGE as the method LLL driven by fpLLL and NTL

[edit] Notes

[edit] References

• Yap, Chee-Keng (2000). "Chap. 8 Gaussian Lattice Reduction - Chap. 9 Lattice Reduction and Applications". Fundamental Problems of Algorithmic Algebra. Oxford, New York: Oxford University Press. pp. 219–257. ISBN 0-19-512516-9. http://www.cs.nyu.edu/yap/book/berlin/. Retrieved 2008-08-25. 

• Borwein, Peter. Computational Excursions in Analysis and Number Theory (ISBN 0-387-95444-9).