lll

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




B

=
{


b


1


,


b


2


,
…
,


b


d


}


{\displaystyle \mathbf {B} =\{\mathbf {b} _{1},\mathbf {b} _{2},\dots ,\mathbf {b} _{d}\}}

with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with



d
≤
n


{\displaystyle d\leq n}

, the LLL algorithm calculates an LLL-reduced (short, nearly orthogonal) lattice basis in time





O


(

d

5


n

log

3


⁡
B
)


{\displaystyle {\mathcal {O}}(d^{5}n\log ^{3}B)}

where



B


{\displaystyle B}

is the largest length of





b


i




{\displaystyle \mathbf {b} _{i}}

under the Euclidean norm, that is,



B
=
max

(

‖


b


1



‖

2


,
‖


b


2



‖

2


,
…
,
‖


b


d



‖

2



)



{\displaystyle B=\max \left(\|\mathbf {b} _{1}\|_{2},\|\mathbf {b} _{2}\|_{2},\dots ,\|\mathbf {b} _{d}\|_{2}\right)}

.
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.

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