Lenstra–Lenstra–Lovász lattice basis reduction algorithm explained

B = \{ b _{1,b} _{2,} ... , 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 $$\backslash mathcal\; O(d^5n\backslash log^3\; B)$$ where B is the largest length of b _{i} under the Euclidean norm, that is, B = max \left(\| b _{1\|} _{2,} \| b _{2\|} _{2,} ... , \| b _{d\|} _{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$$\backslash mathbf=\backslash ,$$define its Gram–Schmidt process orthogonal basis$$\backslash mathbf^*=\backslash ,$$and the Gram-Schmidt coefficients$$\backslash mu\_=\backslash frac,$$ for any

1 \le j < i \le n .

Then the basis

B is LLL-reduced if there exists a parameter \delta in 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.

1. (Lovász condition) For k = 2,3,..,n

\colon \delta \Vert b ^{*} _{k -1} \Vert ^{2} \leq \Vert b ^{2+} _{k\Vert} \mu ^{2 \Vert} _{k,k -1} 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 = (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.^[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.

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

Z ^{4} spanned by [ 1 , 0 , 0 , 10000 r ^{2],} [ 0 , 1 , 0 , 10000 r], 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....

Properties of LLL-reduced basis

Let

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

1. The first vector in the basis cannot be much larger than the shortest non-zero vector:

\Vert b _{1} \Vert \le ( 2 / (\sqrt{ 4 \delta - 1 })) ^{n-1} ⋅ λ _{1(lL)} . In particular, for \delta = 3 / 4 , this gives \Vert b _{1} \Vert \le 2 ^{(n-1)/2} ⋅ λ _{1(lL)} .^[8]

1. The first vector in the basis is also bounded by the determinant of the lattice:

\Vert b _{1} \Vert \le ( 2 / (\sqrt{ 4 \delta - 1 })) ^{(n-1)/2} ⋅ (\det( l L)) ^{1/n} . In particular, for \delta = 3 / 4 , this gives \Vert b _{1} \Vert \le 2 ^{(n-1)/4} ⋅ (\det( l L)) ^{1/n} .

1. The product of the norms of the vectors in the basis cannot be much larger than the determinant of the lattice: let

\delta = 3 / 4 , then $\backslash prod\_^n\; \backslash Vert\backslash mathbf\_i\; \backslash Vert\; \backslash le\; 2^\; \backslash cdot\; \backslash det(\backslash mathcal\; L)$.

LLL algorithm pseudocode

The following description is based on, with the corrections from the errata.^[9]

INPUT a lattice basis b₁, b₂, ..., b_n in Z^m a parameter δ with 1/4 < δ < 1, most commonly δ = 3/4 PROCEDURE B^* <- GramSchmidt = ; and do not normalize μ_i,j <- InnerProduct(b_i, b_j^*)/InnerProduct(b_j^*, b_j^*); using the most current values of b_i and b_j^* k <- 2; while k <= n do for j from k−1 to 1 do if |μ_k,j| > 1/2 then b_k <- b_k − ⌊μ_k,j⌉b_j; Update B^* and the related μ_i,j's as needed. (The naive method is to recompute B^* whenever b_i changes: B^* <- GramSchmidt =) end if end for if InnerProduct(b_k^*, b_k^*) > (δ − μ²_k,k−1) InnerProduct(b_k−1^*, b_k−1^*) then k <- k + 1; else Swap b_k and b_k−1; Update B^* and the related μ_i,j's as needed. k <- max(k−1, 2); end if end while return B the LLL reduced basis of OUTPUT the reduced basis b₁, b₂, ..., b_n in Z^m

Examples

Example from Z³

Let a lattice basis

b _{1,b} _{2,} b _{3} \in Z ^{3} , be given by the columns ofthen the reduced basis is 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 Z[''i'']⁴

Likewise, for the basis over the complex integers given by the columns of the matrix below,then the columns of the matrix below give an LLL-reduced basis.

Implementations

LLL is implemented in

• Arageli as the function lll_reduction_int
• fpLLL as a stand-alone implementation
• FLINT as the function fmpz_lll
• 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

References

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

Notes and References

1. Lenstra. A. K.. A. K. Lenstra. Lenstra. H. W. Jr.. H. W. Lenstra, Jr.. Lovász. L.. László Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen. 261. 1982. 4. 515–534. 1887/3810. 10.1007/BF01457454. 0682664. 10.1.1.310.318. 5701340.
2. Book: Galbraith. Steven. Mathematics of Public Key Cryptography. 2012. chapter 17. https://www.math.auckland.ac.nz/~sgal018/crypto-book/crypto-book.html.
3. Nguyen . Phong Q. . Stehlè . Damien . An LLL Algorithm with Quadratic Complexity . SIAM J. Comput. . September 2009 . 39 . 3 . 874–903 . 10.1137/070705702 . 3 June 2019.
4. Nguyen. Phong Q.. Stehlé. Damien. 1 October 2009. Low-dimensional lattice basis reduction revisited . ACM Transactions on Algorithms . en. 5. 4. 1–48. 10.1145/1597036.1597050. 10583820.
5. Odlyzko . Andrew . te Reile . Herman J. J. . Disproving Mertens Conjecture . . 357 . 138–160 . 10.1515/crll.1985.357.138 . 13016831 . 27 January 2020.
6. D. Wübben et al., "Lattice reduction," IEEE Signal Processing Magazine, Vol. 28, No. 3, pp. 70-91, Apr. 2011.
7. D. Simon . Selected applications of LLL in number theory . LLL+25 Conference . 2007 . Caen, France .
8. Web site: Regev . Oded . Lattices in Computer Science: LLL Algorithm . New York University . 1 February 2019.
9. Web site: Silverman. Joseph. Introduction to Mathematical Cryptography Errata. Brown University Mathematics Dept.. 5 May 2015.
10. Web site: 4. LLL . Bosma. Wieb. Lecture notes. 28 February 2010.
11. A Formalization of the LLL Basis Reduction Algorithm . Divasón. Jose. Conference Paper. Lecture Notes in Computer Science . 2018 . 10895 . 160–177 . 10.1007/978-3-319-94821-8_10 . 978-3-319-94820-1 . free .