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 $\mathbf{B}=\{ \mathbf{b}_1,\mathbf{b}_2, \dots, \mathbf{b}_d \}$ with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rⁿ) with $\ d \leq n$ , the LLL algorithm calculates an LLL-reduced (short, nearly orthogonal) lattice basis in time

O(d^5n\log^3 B)\,

where B is the largest length of $b_i$ 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

\mathbf{B}=\{ \mathbf{b}_0,\mathbf{b}_1, \dots, \mathbf{b}_n \},

define its Gram–Schmidt process orthogonal basis

\mathbf{B}^*=\{ \mathbf{b}^*_0, \mathbf{b}^*_1, \dots, \mathbf{b}^*_n \},

and the Gram-Schmidt coefficients

\mu_{i,j}=\frac{\langle\mathbf{b}_i,\mathbf{b}^*_j\rangle}{\langle\mathbf{b}^*_j,\mathbf{b}^*_j\rangle}

, for any

1 \le j < i \le n

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

(size-reduced) For $1 \leq j < i \leq n\colon \left|\mu_{i,j}\right|\leq 0.5$ . By definition, this property guarantees the length reduction of the ordered basis.
(Lovász condition) For k = 1,2,..,n $\colon \delta \Vert \mathbf{b}^*_{k-1}\Vert^2 \leq \Vert \mathbf{b}^*_k\Vert^2+ \mu_{k,k-1}^2\Vert \mathbf{b}^*_{k-1}\Vert^2$ .

Here, estimating the value of the $\delta$ parameter, we can conclude how well the basis is reduced. Greater values of $\delta$ lead to stronger reductions of the basis. Initially, A. Lenstra, H. Lenstra and L. Lovász demonstrated the LLL-reduction algorithm for $\delta = \frac{3}{4}$ . Note that although LLL-reduction is well-defined for $\delta = 1$ , the polynomial-time complexity is guaranteed only for $\delta$ 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. However, an LLL-reduced basis is nearly as short as possible, in the sense that there are absolute bounds $c_i > 1$ such that the first basis vector is no more than $c_1$ times as long as a shortest vector in the lattice, the second basis vector is likewise within $c_2$ 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:

\triangleright

a lattice basis

b_0, b_1, \dots, b_n \in Z^{m}

\triangleright

parameter

\delta

with

\frac{1}{4} < \delta <1

, most commonly

\delta = \frac{3}{4}

PROCEDURE:

   Perform Gram-Schmidt, but do not normalize:
    $ortho := \mathrm{gramSchmidt}(\{b_0, \dots, b_n\}) = \{ b_0^*, \dots, b_n^* \}$ 
   Define  $\scriptstyle \mu_{i,j}:= \frac{\langle b_{i}, b_{j}^{*} \rangle}{\langle b_{j}^{*}, b_{j}^{*} \rangle}$ , which must always use the most current values of  $\scriptstyle b_i, b_j^*$ .
   Let  $k=1$ 
   while  $k \le n$  do
       for  $j$   from  $k-1$  to  $0$  do
           if  $|\mu_{k,j}| > \frac{1}{2}$  do
                $b_k = b_k - \lfloor \mu_{k,j} \rceil b_j$ 
               Update  $ortho$  entries and related  $\scriptstyle \mu_{i,j}$ 's as needed. 
               (The naive method is to recompute  $\scriptstyle ortho := \mathrm{gramSchmidt}(\{b_0, \dots, b_n\}) = \{ b_0^*, \dots, b_n^* \}$  whenever a  $\scriptstyle b_i$  changes.)
           end if
       end for
       if  $\langle b_k^*, b_k^*\rangle \ge (\delta - (\mu_{k,k-1})^2)\langle b_{k-1}^*, b_{k-1}^*\rangle$  then
            $k = k + 1$ 
       else
           Swap  $\scriptstyle b_k$  and  $\scriptstyle b_{k-1}$ .
           Update  $ortho$  entries and related  $\scriptstyle \mu_{i,j}$ 's as needed. (See above comment.)
            $k = \max(k-1,1)$ 
       end if
   end while

OUTPUT: LLL reduced basis $b_0, b_1, \dots, b_n$

Example

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

INPUT:

Let a lattice basis $\mathbf{b}_1,\mathbf{b}_2, \mathbf{b}_3 \in Z^{3}$ , be given by the columns of

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

Then according to the LLL algorithm we obtain the following:

1. $b_{1}^{*}= b_{1}= \begin{bmatrix}1\\1\\1\end{bmatrix},B_{1}= \langle b_{1}^{*}, b_{1}^{*} \rangle = \begin{bmatrix}1\\1\\1\end{bmatrix} \begin{bmatrix}1\\1\\1\end{bmatrix}= 3$

2.For $i=2$ DO:

2.1.For $j=1$ set $\mu_{2,1}= \frac{\langle b_{2}, b_{1}^{*} \rangle}{B_{1}}= \frac{\begin{bmatrix}-1\\0\\2\end{bmatrix} \begin{bmatrix}1\\1\\1\end{bmatrix}}{3}=\frac{1}{3}(< \frac{1}{2})$

and $b_{2}^{*}= b_{2}- \mu_{2,1}b_{1}^{*}= \begin{bmatrix}-1\\0\\2\end{bmatrix}- \frac{1}{3}\begin{bmatrix}1\\1\\1\end{bmatrix}=\begin{bmatrix}\frac{-4}{3}\\\frac{-1}{3}\\\frac{5}{3}\end{bmatrix}.$

2.2 $B_{2}= \langle b_{2}^{*}, b_{2}^{*} \rangle = \begin{bmatrix}\frac{-4}{3}\\\frac{-1}{3}\\\frac{5}{3}\end{bmatrix} \begin{bmatrix}\frac{-4}{3}\\\frac{-1}{3}\\\frac{5}{3}\end{bmatrix}= \frac{14}{3}.$

3. $\mathbf{k}:=2$

4.Here the step 4 of the LLL algorithm is skipped as size-reduced property holds for $\mu_{2,1}$

5.For $i=3$ and for $j=1,2$ calculate $\mu_{i,j}$ and $B_{i}$ : $\mu_{3,1}= \frac{\langle b_{3}, b_{1}^{*} \rangle}{B_{1}}= \frac{\begin{bmatrix}3\\5\\6\end{bmatrix} \begin{bmatrix}1\\1\\1\end{bmatrix}}{3}=\frac{14}{3}(> \frac{1}{2})$

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

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

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

$B_{3}= \langle b_{3}^{*}, b_{3}^{*} \rangle = \begin{bmatrix}\frac{-6}{14}\\\frac{9}{14}\\\frac{-3}{14}\end{bmatrix} \begin{bmatrix}\frac{-6}{14}\\\frac{9}{14}\\\frac{-3}{14}\end{bmatrix}= \frac{126}{196}= \frac{9}{14}$

6.While $k \leq 3$ DO

6.1 Length reduce $b_{3}$ and correct $\mu_{3,1}$ and $\mu_{3,2}$ according to reduction subroutine in step 4:

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

i. $r = \lfloor 0.5 + \mu_{3,l} \rfloor =5$ and $b_{3} = b_{3}- 5b_{1}= \begin{bmatrix}3\\5\\6\end{bmatrix}- \begin{bmatrix}5\\5\\5\end{bmatrix}=\begin{bmatrix}-2\\0\\1\end{bmatrix}$

ii. $\mu_{3,1}= \mu_{3,l} - r\mu_{1,1} = \frac{-1}{3}(< \frac{1}{2})$

iii.Set $\mu_{3,1}= \mu_{3,1} - r= \frac{14}{3}-5= \frac{-1}{3}$

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

i. $r = \lfloor 0.5 + \mu_{3,2} \rfloor =1$ and $b_{3} = b_{3}- b_{2}= \begin{bmatrix}3\\5\\6\end{bmatrix}- \begin{bmatrix}-1\\0\\2\end{bmatrix}=\begin{bmatrix}4\\5\\4\end{bmatrix}$

ii.Set $\mu_{3,2}= \mu_{3,2} - r\mu_{2,2}= \frac{13}{14}-1= \frac{-1}{14}$

iii. $\mu_{3,2}= \mu_{3,2} - 1 = \frac{-1}{14}(< \frac{1}{2})$

6.2 As $B_{3} < (\frac{3}{4}- \mu_{3,2}^2)B_{2}$ takes place, then

6.2.1 Exchange $b_{3}$ and $b_{2}$

6.2.2 $k$ := 2

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

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

Implement the algorithm steps again. 1. $b_{1}^{*}= b_{1}= \begin{bmatrix}1\\1\\1\end{bmatrix},B_{1}= 3$

2. $\mu_{2,1}= \frac{\langle b_{2}, b_{1}^{*} \rangle}{B_{1}}= \frac{\begin{bmatrix}4\\5\\4\end{bmatrix} \begin{bmatrix}1\\1\\1\end{bmatrix}}{3}=\frac{13}{3}(>\frac{1}{2})$

3. $b_{2}^{*}= b_{2}- \mu_{2,1}b_{1}^{*}= \begin{bmatrix}4\\5\\4\end{bmatrix}- \frac{13}{3}\begin{bmatrix}1\\1\\1\end{bmatrix}=\begin{bmatrix}\frac{-1}{3}\\\frac{2}{3}\\\frac{-1}{3}\end{bmatrix}$ .

4. $B_{2}= \langle b_{2}^{*}, b_{2}^{*} \rangle = \frac{2}{3}$ .

5.For $\mid \mu_{2,1}\mid >\frac{1}{2}$ EXECUTE reduction subroutine RED(2,1):

i. $r = \lfloor 0.5 + \mu_{2,l} \rfloor =4$ and $b_{2} = b_{2}- 4b_{1}= \begin{bmatrix}4\\5\\4\end{bmatrix}- \begin{bmatrix}4\\4\\4\end{bmatrix}=\begin{bmatrix}0\\1\\0\end{bmatrix}$

ii.Set $\mu_{2,1}= \mu_{2,1} - 4\mu_{1,1}= \frac{13}{3}- 4= \frac{1}{3}(< \frac{1}{2})$

6. As $B_{2} < (\frac{3}{4}- \mu_{2,1}^2)B_{1}$ takes place, then

7. Exchange $b_{2}$ and $b_{1}$

OUTPUT: LLL reduced basis

\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 $R^4$ spanned by $[1,0,0,10000r^2], [0,1,0,10000r],$ 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 $[a,b,c,10000(ar^2+br+c)]$ ; but such a vector is "short" only if a, b, c are small and $ar^2+br+c$ 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 $x^2-x-1$ 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
Sage as the method LLL driven by fpLLL and NTL

Notes

↑ Lenstra, A. K.; Lenstra, H. W., Jr.; Lovász, L. (1982). "Factoring polynomials with rational coefficients". Mathematische Annalen 261 (4): 515–534. doi:10.1007/BF01457454. MR 0682664. hdl:1887/3810.
↑ Silverman, Joseph. "Introduction to Mathematical Cryptography Errata" (PDF). Brown University Mathematics Dept. Retrieved 5 May 2015.
↑ Bosma, Wieb. "4. LLL" (PDF). Lecture notes. Retrieved 28 February 2010.
↑ Shahabuddin, Shahriar et al., "A Customized Lattice Reduction Multiprocessor for MIMO Detection", in Arxiv preprint, January 2015.
↑ D. Simon (2007). "Selected applications of LLL in number theory" (PDF). LLL+25 Conference (Caen, France).

References

Napias, Huguette (1996). "A generalizaion of the LLL algorithm over euclidean rings or orders". J. The. Nombr. Bordeaux 8 (2): 387–396.
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". Lin. Alg. Appl. 434: 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.

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