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 with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with , the LLL algorithm calculates an LLL-reduced (short, nearly orthogonal) lattice basis in time

where B is the largest length of 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

define its Gram–Schmidt process orthogonal basis

and the Gram-Schmidt coefficients

, for any .

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

  1. (size-reduced) For . By definition, this property guarantees the length reduction of the ordered basis.
  2. (Lovász condition) For k = 1,2,..,n .

Here, estimating the value of the parameter, we can conclude how well the basis is reduced. Greater values of lead to stronger reductions of the basis. Initially, A. Lenstra, H. Lenstra and L. Lovász demonstrated the LLL-reduction algorithm for . Note that although LLL-reduction is well-defined for , the polynomial-time complexity is guaranteed only for 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 such that the first basis vector is no more than times as long as a shortest vector in the lattice, the second basis vector is likewise within of the second successive minimum, and so on.

LLL Algorithm[edit]

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

INPUT:

a lattice basis ,
parameter with , most commonly

PROCEDURE:

   Perform Gram-Schmidt, but do not normalize:
   
   Define , which must always use the most current values of .
   Let 
   while  do
       for j  from  to 0 do
           if  do
               
               Update ortho entries and related 's as needed. 
               (The naive method is to recompute  whenever a  changes.)
           end if
       end for
       if  then
           
       else
           Swap  and .
           Update ortho entries and related 's as needed. (See above comment.)
           
       end if
   end while

OUTPUT: LLL reduced basis

Example[edit]

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

INPUT:

Let a lattice basis , be given by the columns of

Then according to the LLL algorithm we obtain the following:

  1. For DO:
    1. For set and
  2. Here the step 4 of the LLL algorithm is skipped as size-reduced property holds for
  3. For and for calculate and :

    hence

    and

    hence and

  4. While DO
    1. Length reduce and correct and

      according to reduction subroutine in step 4: For EXECUTE reduction subroutine RED(3,1):

      1. and
      2. Set

      For EXECUTE reduction subroutine RED(3,2):

      1. and
      2. Set
    2. As takes place, then
      1. Exchange and

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

Implement the algorithm steps again.

  1. .
  2. .
  3. For EXECUTE reduction subroutine RED(2,1):
    1. and
    2. Set
  4. As takes place, then
  5. Exchange and
OUTPUT:

LLL reduced basis

Applications[edit]

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 spanned by and . The first vector in the reduced basis will be an integer linear combination of these three, thus necessarily of the form ; but such a vector is "short" only if a, b, c are small and 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 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. ^ Silverman, Joseph. "Introduction to Mathematical Cryptography Errata" (PDF). Brown University Mathematics Dept. Retrieved 5 May 2015. 
  3. ^ Bosma, Wieb. "4. LLL" (PDF). Lecture notes. Retrieved 28 February 2010. 
  4. ^ Shahabuddin, Shahriar et al., "A Customized Lattice Reduction Multiprocessor for MIMO Detection", in Arxiv preprint, January 2015.
  5. ^ D. Simon (2007). "Selected applications of LLL in number theory" (PDF). LLL+25 Conference. Caen, France. 

References[edit]