Lenstra–Lenstra–Lovász lattice basis reduction 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 ${\displaystyle B}$ is the largest length of ${\displaystyle \mathbf {b} _{i}}$ under the Euclidean norm, that is, ${\displaystyle B=\max \left(\Vert \mathbf {b} _{1}\Vert _{2},\Vert \mathbf {b} _{2}\Vert _{2},...,\Vert \mathbf {b} _{d}\Vert _{2}\right)}$.^[2][3]

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

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

define its Gram–Schmidt process orthogonal basis

${\displaystyle \mathbf {B} ^{*}=\{\mathbf {b} _{1}^{*},\mathbf {b} _{2}^{*},\dots ,\mathbf {b} _{n}^{*}\},}$

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 ${\displaystyle 1\leq j<i\leq n}$.

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 ${\displaystyle 1\leq j<i\leq n\colon \left|\mu _{i,j}\right|\leq 0.5}$. By definition, this property guarantees the length reduction of the ordered basis.
2. (Lovász condition) For k = 2,3,..,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 ${\displaystyle (0.25,1)}$.

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.^[4] 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.

Applications[]

An early successful application of the LLL algorithm was its use by Andrew Odlyzko and Herman te Riele in disproving Mertens conjecture.^[5]

The LLL algorithm has found numerous other applications in MIMO detection algorithms^[6] 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.^[7]

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 ${\displaystyle \mathbf {Z} ^{4}}$ 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....

Properties of LLL-reduced basis[]

Let ${\displaystyle \mathbf {B} =\{\mathbf {b} _{1},\mathbf {b} _{2},\dots ,\mathbf {b} _{n}\}}$ be a ${\displaystyle \delta }$-LLL-reduced basis of a lattice ${\displaystyle {\mathcal {L}}}$. From the definition of LLL-reduced basis, we can derive several other useful properties about ${\displaystyle \mathbf {B} }$.

1. The first vector in the basis cannot be much larger than the shortest non-zero vector: ${\displaystyle \Vert \mathbf {b} _{1}\Vert \leq (2/({\sqrt {4\delta -1}}))^{n-1}\cdot \lambda _{1}({\mathcal {L}})}$. In particular, for ${\displaystyle \delta =3/4}$, this gives ${\displaystyle \Vert \mathbf {b} _{1}\Vert \leq 2^{(n-1)/2}\cdot \lambda _{1}({\mathcal {L}})}$.^[8]
2. The first vector in the basis is also bounded by the determinant of the lattice: ${\displaystyle \Vert \mathbf {b} _{1}\Vert \leq (2/({\sqrt {4\delta -1}}))^{(n-1)/2}\cdot (\det({\mathcal {L}}))^{1/n}}$. In particular, for ${\displaystyle \delta =3/4}$, this gives ${\displaystyle \Vert \mathbf {b} _{1}\Vert \leq 2^{(n-1)/4}\cdot (\det({\mathcal {L}}))^{1/n}}$.
3. The product of the norms of the vectors in the basis cannot be much larger than the determinant of the lattice: let ${\displaystyle \delta =3/4}$, then ${\displaystyle \prod _{i=1}^{n}\Vert \mathbf {b} _{i}\Vert \leq 2^{n(n-1)/4}\cdot \det({\mathcal {L}})}$.

LLL algorithm pseudocode[]

The following description is based on (Hoffstein, Pipher & Silverman 2008, Theorem 6.68), with the corrections from the errata.^[9]

INPUT
    a lattice basis ${\displaystyle \mathbf {b} _{0},\mathbf {b} _{1},\ldots ,\mathbf {b} _{n}\in \mathbb {Z} ^{m}}$
    a parameter ${\displaystyle \delta }$ with ${\displaystyle {\tfrac {1}{4}}<\delta <1}$, most commonly ${\displaystyle \delta ={\tfrac {3}{4}}}$
PROCEDURE
    ${\displaystyle \mathbf {B^{\ast }} \gets {\rm {GramSchmidt}}(\{\mathbf {b} _{0},\ldots ,\mathbf {b} _{n}\})=\{\mathbf {b} _{0}^{\ast },\ldots ,\mathbf {b} _{n}^{\ast }\};}$  and do not normalize
    ${\displaystyle \mu _{i,j}\gets {\frac {\langle \mathbf {b} _{i},\mathbf {b} _{j}^{\ast }\rangle }{\langle \mathbf {b} _{j}^{\ast },\mathbf {b} _{j}^{\ast }\rangle }};}$   using the most current values of ${\displaystyle \mathbf {b} _{i}}$ and ${\displaystyle \mathbf {b} _{j}^{\ast }}$
    ${\displaystyle k\gets 1;}$
    while ${\displaystyle k\leq n}$ do
        for ${\displaystyle j}$ from ${\displaystyle k-1}$ to ${\displaystyle 0}$ do
            if ${\displaystyle |\mu _{k,j}|>{\tfrac {1}{2}}}$ then
                ${\displaystyle \mathbf {b} _{k}\gets \mathbf {b} _{k}-\lfloor \mu _{k,j}\rceil \mathbf {b} _{j};}$
               Update ${\displaystyle \mathbf {B^{\ast }} }$ and the related ${\displaystyle \mu _{i,j}}$'s as needed.
               (The naive method is to recompute ${\displaystyle \mathbf {B^{\ast }} }$ whenever ${\displaystyle \mathbf {b} _{i}}$ changes:
                ${\displaystyle \mathbf {B^{\ast }} \gets {\rm {GramSchmidt}}(\{\mathbf {b} _{0},\ldots ,\mathbf {b} _{n}\})=\{\mathbf {b} _{0}^{\ast },\ldots ,\mathbf {b} _{n}^{\ast }\};}$)
            end if
        end for
        if ${\displaystyle \langle \mathbf {b} _{k}^{\ast },\mathbf {b} _{k}^{\ast }\rangle \geq \left(\delta -\mu _{k,k-1}^{2}\right)\langle \mathbf {b} _{k-1}^{\ast },\mathbf {b} _{k-1}^{\ast }\rangle }$ then
            ${\displaystyle k\gets k+1;}$
        else
            Swap ${\displaystyle \mathbf {b} _{k}}$ and  ${\displaystyle \mathbf {b} _{k-1};}$
            Update ${\displaystyle \mathbf {B^{\ast }} }$ and the related ${\displaystyle \mu _{i,j}}$'s as needed.
            ${\displaystyle k\gets \max(k-1,1);}$
        end if
    end while
    return ${\displaystyle \mathbf {B} }$ the LLL reduced basis of ${\displaystyle \{b_{0},\ldots ,b_{n}\}}$
OUTPUT
    the reduced basis ${\displaystyle \mathbf {b} _{0},\mathbf {b} _{1},\ldots ,\mathbf {b} _{n}\in \mathbb {Z} ^{m}}$

Examples[]

Example from ${\displaystyle \mathbf {Z} ^{3}}$[]

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

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

then the reduced basis is

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

which is size-reduced, satisfies the Lovász condition, and is hence LLL-reduced, as described above. See W. Bosma.^[10] for details of the reduction process.

Example from ${\displaystyle \mathbf {Z} [i]^{4}}$[]

Likewise, for the basis over the complex integers given by the columns of the matrix below,

${\displaystyle {\begin{bmatrix}-2+2i&7+3i&7+3i&-5+4i\\3+3i&-2+4i&6+2i&-1+4i\\2+2i&-8+0i&-9+1i&-7+5i\\8+2i&-9+0i&6+3i&-4+4i\end{bmatrix}}}$,

then the columns of the matrix below give an LLL-reduced basis.

${\displaystyle {\begin{bmatrix}-6+3i&-2+2i&2-2i&-3+6i\\6-1i&3+3i&5-5i&2+1i\\2-2i&2+2i&-3-1i&-5+3i\\-2+1i&8+2i&7+1i&-2-4i\\\end{bmatrix}}}$.

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
• Isabelle/HOL in the 'archive of formal proofs' entry LLL_Basis_Reduction. This code exports to efficiently executable Haskell.^[11]

See also[]

• Coppersmith method

Notes[]

References[]

• Napias, Huguette (1996). "A generalization of the LLL algorithm over euclidean rings or orders". Journal de Théorie des Nombres de Bordeaux. 8 (2): 387–396. doi:10.5802/jtnb.176.
• Cohen, Henri (2000). A course in computational algebraic number theory. GTM. 138. Springer. ISBN 3-540-55640-0.
• Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. ISBN 0-387-95444-9.
• Luk, Franklin T.; Qiao, Sanzheng (2011). "A pivoted LLL algorithm". Linear Algebra and Its Applications. 434 (11): 2296–2307. doi:10.1016/j.laa.2010.04.003.
• Hoffstein, Jeffrey; Pipher, Jill; Silverman, J.H. (2008). An Introduction to Mathematical Cryptography. Springer. ISBN 978-0-387-77993-5.