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 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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