Lenstra–Lenstra–Lovász lattice basis reduction algorithm

(Redirected from LLL algorithm)

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.[1] Given a basis ${\displaystyle \mathbf {B} =\{\mathbf {b} _{1},\mathbf {b} _{2},\dots ,\mathbf {b} _{d}\}}$ with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with ${\displaystyle \ d\leq n}$, the LLL algorithm calculates an LLL-reduced (short, nearly orthogonal) lattice basis in time

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

where B is the largest length of ${\displaystyle b_{i}}$ under the Euclidean norm.

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

${\displaystyle \mathbf {B} =\{\mathbf {b} _{0},\mathbf {b} _{1},\dots ,\mathbf {b} _{n}\},}$

define its Gram–Schmidt process orthogonal basis

${\displaystyle \mathbf {B} ^{*}=\{\mathbf {b} _{0}^{*},\mathbf {b} _{1}^{*},\dots ,\mathbf {b} _{n}^{*}\},}$

and the Gram-Schmidt coefficients

${\displaystyle \mu _{i,j}={\frac {\langle \mathbf {b} _{i},\mathbf {b} _{j}^{*}\rangle }{\langle \mathbf {b} _{j}^{*},\mathbf {b} _{j}^{*}\rangle }}}$, for any ${\displaystyle 1\leq j.

Then the basis ${\displaystyle B}$ is LLL-reduced if there exists a parameter ${\displaystyle \delta }$ in (0.25,1] such that the following holds:

1. (size-reduced) For ${\displaystyle 1\leq j. By definition, this property guarantees the length reduction of the ordered basis.
2. (Lovász condition) For k = 1,2,..,n ${\displaystyle \colon \delta \Vert \mathbf {b} _{k-1}^{*}\Vert ^{2}\leq \Vert \mathbf {b} _{k}^{*}\Vert ^{2}+\mu _{k,k-1}^{2}\Vert \mathbf {b} _{k-1}^{*}\Vert ^{2}}$.

Here, estimating the value of the ${\displaystyle \delta }$ parameter, we can conclude how well the basis is reduced. Greater values of ${\displaystyle \delta }$ lead to stronger reductions of the basis. Initially, A. Lenstra, H. Lenstra and L. Lovász demonstrated the LLL-reduction algorithm for ${\displaystyle \delta ={\frac {3}{4}}}$. Note that although LLL-reduction is well-defined for ${\displaystyle \delta =1}$, the polynomial-time complexity is guaranteed only for ${\displaystyle \delta }$ 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.[citation needed] However, an LLL-reduced basis is nearly as short as possible, in the sense that there are absolute bounds ${\displaystyle c_{i}>1}$ such that the first basis vector is no more than ${\displaystyle c_{1}}$ times as long as a shortest vector in the lattice, the second basis vector is likewise within ${\displaystyle c_{2}}$ of the second successive minimum, and so on.

LLL Algorithm

The following description is based on (Hoffstein, Pipher & Silverman 2008, Theorem 6.68), with the corrections from the errata [2]

INPUT:

${\displaystyle \triangleright }$ a lattice basis ${\displaystyle b_{0},b_{1},\dots ,b_{n}\in Z^{m}}$,
${\displaystyle \triangleright }$ parameter ${\displaystyle \delta }$ with ${\displaystyle {\frac {1}{4}}<\delta <1}$, most commonly ${\displaystyle \delta ={\frac {3}{4}}}$

PROCEDURE:

   Perform Gram-Schmidt, but do not normalize:
${\displaystyle ortho:=\mathrm {gramSchmidt} (\{b_{0},\dots ,b_{n}\})=\{b_{0}^{*},\dots ,b_{n}^{*}\}}$
Define ${\displaystyle \scriptstyle \mu _{i,j}:={\frac {\langle b_{i},b_{j}^{*}\rangle }{\langle b_{j}^{*},b_{j}^{*}\rangle }}}$, which must always use the most current values of ${\displaystyle \scriptstyle b_{i},b_{j}^{*}}$.
Let ${\displaystyle k=1}$
while ${\displaystyle k\leq n}$ do
for j  from ${\displaystyle k-1}$ to 0 do
if ${\displaystyle |\mu _{k,j}|>{\frac {1}{2}}}$ do
${\displaystyle b_{k}=b_{k}-\lfloor \mu _{k,j}\rceil b_{j}}$
Update ortho entries and related ${\displaystyle \scriptstyle \mu _{i,j}}$'s as needed.
(The naive method is to recompute ${\displaystyle \scriptstyle ortho:=\mathrm {gramSchmidt} (\{b_{0},\dots ,b_{n}\})=\{b_{0}^{*},\dots ,b_{n}^{*}\}}$ whenever a ${\displaystyle \scriptstyle b_{i}}$ changes.)
end if
end for
if ${\displaystyle \langle b_{k}^{*},b_{k}^{*}\rangle \geq (\delta -(\mu _{k,k-1})^{2})\langle b_{k-1}^{*},b_{k-1}^{*}\rangle }$ then
${\displaystyle k=k+1}$
else
Swap ${\displaystyle \scriptstyle b_{k}}$ and ${\displaystyle \scriptstyle b_{k-1}}$.
Update ortho entries and related ${\displaystyle \scriptstyle \mu _{i,j}}$'s as needed. (See above comment.)
${\displaystyle k=\max(k-1,1)}$
end if
end while


OUTPUT: LLL reduced basis ${\displaystyle b_{0},b_{1},\dots ,b_{n}}$

Example

The following presents an example due to W. Bosma.[3]

INPUT:

Let a lattice basis ${\displaystyle \mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\in Z^{3}}$, be given by the columns of

${\displaystyle {\begin{bmatrix}1&-1&3\\1&0&5\\1&2&6\end{bmatrix}}}$

Then according to the LLL algorithm we obtain the following:

1. ${\displaystyle b_{1}^{*}=b_{1}={\begin{bmatrix}1\\1\\1\end{bmatrix}},B_{1}=\langle b_{1}^{*},b_{1}^{*}\rangle ={\begin{bmatrix}1\\1\\1\end{bmatrix}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}=3}$
2. For ${\displaystyle i=2}$ DO:
1. For ${\displaystyle j=1}$ set ${\displaystyle \mu _{2,1}={\frac {\langle b_{2},b_{1}^{*}\rangle }{B_{1}}}={\frac {{\begin{bmatrix}-1\\0\\2\end{bmatrix}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}}{3}}={\frac {1}{3}}\left(<{\frac {1}{2}}\right)}$ and ${\displaystyle b_{2}^{*}=b_{2}-\mu _{2,1}b_{1}^{*}={\begin{bmatrix}-1\\0\\2\end{bmatrix}}-{\frac {1}{3}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}={\begin{bmatrix}{\frac {-4}{3}}\\{\frac {-1}{3}}\\{\frac {5}{3}}\end{bmatrix}}.}$
2. ${\displaystyle B_{2}=\langle b_{2}^{*},b_{2}^{*}\rangle ={\begin{bmatrix}{\frac {-4}{3}}\\{\frac {-1}{3}}\\{\frac {5}{3}}\end{bmatrix}}{\begin{bmatrix}{\frac {-4}{3}}\\{\frac {-1}{3}}\\{\frac {5}{3}}\end{bmatrix}}={\frac {14}{3}}.}$
3. ${\displaystyle \mathbf {k} :=2}$
4. Here the step 4 of the LLL algorithm is skipped as size-reduced property holds for ${\displaystyle \mu _{2,1}}$
5. For ${\displaystyle i=3}$ and for ${\displaystyle j=1,2}$ calculate ${\displaystyle \mu _{i,j}}$ and ${\displaystyle B_{i}}$:

${\displaystyle \mu _{3,1}={\frac {\langle b_{3},b_{1}^{*}\rangle }{B_{1}}}={\frac {{\begin{bmatrix}3\\5\\6\end{bmatrix}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}}{3}}={\frac {14}{3}}(>{\frac {1}{2}})}$

hence ${\displaystyle b_{3}^{*}=b_{3}-\mu _{3,1}b_{1}^{*}={\begin{bmatrix}3\\5\\6\end{bmatrix}}-{\frac {14}{3}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}={\begin{bmatrix}{\frac {-5}{3}}\\{\frac {1}{3}}\\{\frac {4}{3}}\end{bmatrix}}}$

and ${\displaystyle \mu _{3,2}={\frac {\langle b_{3},b_{2}^{*}\rangle }{B_{2}}}={\frac {{\begin{bmatrix}3\\5\\6\end{bmatrix}}{\begin{bmatrix}{\frac {-4}{3}}\\{\frac {-1}{3}}\\{\frac {5}{3}}\end{bmatrix}}}{\frac {14}{3}}}={\frac {13}{14}}\left(>{\frac {1}{2}}\right)}$

hence ${\displaystyle b_{3}^{*}=b_{3}^{*}-\mu _{3,2}b_{2}^{*}={\begin{bmatrix}{\frac {-5}{3}}\\{\frac {1}{3}}\\{\frac {4}{3}}\end{bmatrix}}-{\frac {13}{14}}{\begin{bmatrix}{\frac {-4}{3}}\\{\frac {-1}{3}}\\{\frac {5}{3}}\end{bmatrix}}={\begin{bmatrix}{\frac {-18}{42}}\\{\frac {27}{42}}\\{\frac {-9}{42}}\end{bmatrix}}={\begin{bmatrix}{\frac {-6}{14}}\\{\frac {9}{14}}\\{\frac {-3}{14}}\end{bmatrix}}}$ and

${\displaystyle B_{3}=\langle b_{3}^{*},b_{3}^{*}\rangle ={\begin{bmatrix}{\frac {-6}{14}}\\{\frac {9}{14}}\\{\frac {-3}{14}}\end{bmatrix}}{\begin{bmatrix}{\frac {-6}{14}}\\{\frac {9}{14}}\\{\frac {-3}{14}}\end{bmatrix}}={\frac {126}{196}}={\frac {9}{14}}}$
6. While ${\displaystyle k\leq 3}$ DO
1. Length reduce ${\displaystyle b_{3}}$ and correct ${\displaystyle \mu _{3,1}}$ and ${\displaystyle \mu _{3,2}}$

according to reduction subroutine in step 4: For ${\displaystyle \mid \mu _{3,1}\mid >{\frac {1}{2}}}$ EXECUTE reduction subroutine RED(3,1):

1. ${\displaystyle r=\lfloor 0.5+\mu _{3,l}\rfloor =5}$ and ${\displaystyle b_{3}=b_{3}-5b_{1}={\begin{bmatrix}3\\5\\6\end{bmatrix}}-{\begin{bmatrix}5\\5\\5\end{bmatrix}}={\begin{bmatrix}-2\\0\\1\end{bmatrix}}}$
2. ${\displaystyle \mu _{3,1}=\mu _{3,l}-r\mu _{1,1}={\frac {-1}{3}}\left(<{\frac {1}{2}}\right)}$
3. Set ${\displaystyle \mu _{3,1}=\mu _{3,1}-r={\frac {14}{3}}-5={\frac {-1}{3}}}$

For ${\displaystyle \mid \mu _{3,2}\mid >{\frac {1}{2}}}$ EXECUTE reduction subroutine RED(3,2):

1. ${\displaystyle r=\lfloor 0.5+\mu _{3,2}\rfloor =1}$ and ${\displaystyle b_{3}=b_{3}-b_{2}={\begin{bmatrix}3\\5\\6\end{bmatrix}}-{\begin{bmatrix}-1\\0\\2\end{bmatrix}}={\begin{bmatrix}4\\5\\4\end{bmatrix}}}$
2. Set ${\displaystyle \mu _{3,2}=\mu _{3,2}-r\mu _{2,2}={\frac {13}{14}}-1={\frac {-1}{14}}}$
3. ${\displaystyle \mu _{3,2}=\mu _{3,2}-1={\frac {-1}{14}}\left(<{\frac {1}{2}}\right)}$
2. As ${\displaystyle B_{3}<\left({\frac {3}{4}}-\mu _{3,2}^{2}\right)B_{2}}$ takes place, then
1. Exchange ${\displaystyle b_{3}}$ and ${\displaystyle b_{2}}$
2. ${\displaystyle k:=2}$

Apply a SWAP, continue algorithm with the lattice basis, which is given by columns

${\displaystyle {\begin{bmatrix}1&4&-1\\1&5&0\\1&4&2\end{bmatrix}}}$

Implement the algorithm steps again.

1. ${\displaystyle b_{1}^{*}=b_{1}={\begin{bmatrix}1\\1\\1\end{bmatrix}},B_{1}=3}$
2. ${\displaystyle \mu _{2,1}={\frac {\langle b_{2},b_{1}^{*}\rangle }{B_{1}}}={\frac {{\begin{bmatrix}4\\5\\4\end{bmatrix}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}}{3}}={\frac {13}{3}}\left(>{\frac {1}{2}}\right)}$
3. ${\displaystyle b_{2}^{*}=b_{2}-\mu _{2,1}b_{1}^{*}={\begin{bmatrix}4\\5\\4\end{bmatrix}}-{\frac {13}{3}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}={\begin{bmatrix}{\frac {-1}{3}}\\{\frac {2}{3}}\\{\frac {-1}{3}}\end{bmatrix}}}$.
4. ${\displaystyle B_{2}=\langle b_{2}^{*},b_{2}^{*}\rangle ={\frac {2}{3}}}$.
5. For ${\displaystyle \mid \mu _{2,1}\mid >{\frac {1}{2}}}$ EXECUTE reduction subroutine RED(2,1):
1. ${\displaystyle r=\lfloor 0.5+\mu _{2,l}\rfloor =4}$ and ${\displaystyle b_{2}=b_{2}-4b_{1}={\begin{bmatrix}4\\5\\4\end{bmatrix}}-{\begin{bmatrix}4\\4\\4\end{bmatrix}}={\begin{bmatrix}0\\1\\0\end{bmatrix}}}$
2. Set ${\displaystyle \mu _{2,1}=\mu _{2,1}-4\mu _{1,1}={\frac {13}{3}}-4={\frac {1}{3}}\left(<{\frac {1}{2}}\right)}$
6. As ${\displaystyle B_{2}<\left({\frac {3}{4}}-\mu _{2,1}^{2}\right)B_{1}}$ takes place, then
7. Exchange ${\displaystyle b_{2}}$ and ${\displaystyle b_{1}}$
OUTPUT:

LLL reduced basis

${\displaystyle {\begin{bmatrix}0&1&-1\\1&0&0\\0&1&2\end{bmatrix}}}$

Applications

The LLL algorithm has found numerous other applications in MIMO detection algorithms [4] 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.[5]

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 ${\displaystyle R^{4}}$ spanned by ${\displaystyle [1,0,0,10000r^{2}],[0,1,0,10000r],}$ and ${\displaystyle [0,0,1,10000]}$. The first vector in the reduced basis will be an integer linear combination of these three, thus necessarily of the form ${\displaystyle [a,b,c,10000(ar^{2}+br+c)]}$; but such a vector is "short" only if a, b, c are small and ${\displaystyle ar^{2}+br+c}$ 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 ${\displaystyle x^{2}-x-1}$ has a root equal to the golden ratio, 1.6180339887….

Implementations

LLL is implemented in

• Arageli as the function lll_reduction_int
• fpLLL as a stand-alone implementation
• GAP as the function LLLReducedBasis
• 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
• Pymatgen as the function analysis.get_lll_reduced_lattice
• SageMath as the method LLL driven by fpLLL and NTL