Lenstra–Lenstra–Lovász lattice basis reduction algorithm

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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 in Rn with  \ d \leq n , the LLL algorithm outputs 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[edit]

The precise definition of LLL-reduced is as follows: Given a basis

\mathbf{B}=\{ \mathbf{b}_1,\mathbf{b}_2, \dots, \mathbf{b}_n \},

define its Gram–Schmidt process orthogonal basis

\mathbf{B}^*=\{ \mathbf{b}^*_1, \mathbf{b}^*_2, \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:

  1. (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.
  2. (Lovász condition) For k = 2,3,..,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[edit]

The following description is based on (Cohen 2000, Algorithm 2.6.3), but currently is incorrect.

INPUT:

\triangleright a lattice basis  \mathbf{b}_1,\mathbf{b}_2, \dots, \mathbf{b}_n  \in Z^{m},
\triangleright parameter \delta with \frac{1}{4} < \delta <1

PROCEDURE:

Perform Gram-Schmidt:

  • b_{1}^{*}:= b_{1},B_{1}:= \langle b_{1}^{*}, b_{1}^{*} \rangle
  • for i from 2 to n do
    • b_{i}^{*}:= b_{i}
    • for j from 1 to i-1 do
      • \mu_{i,j}:= \frac{\langle b_{i}, b_{j}^{*} \rangle}{B_{j}}
      • b_{i}^{*}:= b_{i}^{*} - \mu_{i,j}b_{j}^{*}
    • end for
    • B_{i}:= \langle b_{i}^{*}, b_{i}^{*} \rangle
  • end for
  • k:=2 (k is the stage at which the vectors  \mathbf{b}_1,\mathbf{b}_2, \dots, \mathbf{b}_{k-1} are reduced according to size-reduced property 1.)
  • if |\mu_{i,j}| >\frac{1}{2} then execute reduction subroutine RED(k,k-1):
    • for l from k-1 to 1 do
      • r:= \lfloor 0.5 + \mu_{k,l} \rfloor
      • b_{k}:= b_{k}- rb_{l}
      • for j from 1 to l-1 do
        •  \mu_{k,j}:= \mu_{k,j} - r \mu_{l,j}
      • end for
      •  \mu_{k,l}:= \mu_{k,l} - r
    • end for
  • end if
  • Calculate  \mu_{i,j} for 1 \leq j <i \leq n and B_{i} for i from 1 to n
  • while k \leq n do
    • Length reduce b_{k} and correct \mu_{k,j} according to reduction subroutine in step 4, for j from 1 till k-1
    • if  B_{k} < (\frac{3}{4}- \mu_{k,k-1}^2)B_{k-1} then
      • Exchange b_{k} and b_{k-1}
      • k:= max (2,k-1)
    • else
      • k:= k+1
    • end if
  • end while

OUTPUT: LLL reduced basis  \mathbf{b}_1,\mathbf{b}_2, \dots, \mathbf{b}_n

Example[edit]

The following presents an example due to W. Bosma.[2]

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}.

2.2B_{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[edit]

The LLL algorithm has found numerous other applications in MIMO detection algorithms 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.[3]

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 a quadratic equation with integer coefficients, one may apply the 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[edit]

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

See also[edit]

Notes[edit]

  1. ^ 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. 
  2. ^ Bosma, Wieb. "4. LLL". Lecture notes. Retrieved 28 February 2010. 
  3. ^ D. Simon (2007). "Selected applications of LLL in number theory". LLL+25 Conference (Caen, France). 

References[edit]