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= The Road to LLL in SAGE =
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== LLL Heuristic ==
To develop a simple heuristic how to choose a LLL implementation, we thought about using the following benchmarking examples. All these examples are generated using Stehle's {{{generate.c}}}} code and follow his slides for dimensions and bitsizes.

 * 1000 dimensional matrices filled uniformly random with integers of 10, 100, or 1000 bits respectively.

 * matrices as they occur for the Knapsack problem with (dimension,bitsize) pairs of (10, 100000), (100,10000), (150,5000)

 * matrices as they appear for solving simultaneous Diophantine equations of (dim,bits) pairs (3, 128), (12, 10000), (76, 5000)

 * Ajtai (d, bits) (10, 7), (2, 13), (3, 11)

 * particular bad matrices with entries sized at 64, 128, and 500 bits.

 * NTRU (dim, bits, q) (10,100,12), (100,100,35), (5,1000,75)

The Road to LLL in SAGE

[http://perso.ens-lyon.fr/damien.stehle/english.html Damien Stehle]'s fpLLL code is wrapped at [http://trac.sagemath.org/sage_trac/ticket/723 #723] or [http://sage.math.washington.edu/home/malb/fplll.patch fplll.patch] respectively. For some problems, this gives quite good performance already:

   1 sage: B = MatrixSpace(IntegerRing(), 50, 51)(0)
   2 sage: for i in range(50): B[i,0] = ZZ.random_element(2**10000)
   3 ....:
   4 sage: for i in range(50): B[i,i+1] = 1
   5 ....:
   6 sage: time C = B.LLL('fpLLL:fast')
   7 CPU times: user 9.54 s, sys: 0.00 s, total: 9.54 s
   8 Wall time: 9.56
   9 
  10 sage: C.is_LLL_reduced()
  11 True
  12 
  13 sage: BM = B._magma_()
  14 sage: time CM = BM.LLL()
  15 CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
  16 Wall time: 15.20
  17 
  18 sage: time magma.eval("CM := LLL(%s:Fast:=1)"%BM.name())
  19 CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
  20 Wall time: 11.68

However, the main strength of MAGMA's LLL is that it chooses a reasonably 'good' LLL implementation automatically. This is described in Damien Stehle's [http://perso.ens-lyon.fr/damien.stehle/FPLLL_SURVEY.html paper] and timings can be found in some of his [http://magma.maths.usyd.edu.au/Magma2006/talks/stehle.pdf slides]. For those examples SAGE seems to perform quite poorly.

LLL Heuristic

To develop a simple heuristic how to choose a LLL implementation, we thought about using the following benchmarking examples. All these examples are generated using Stehle's generate.c} code and follow his slides for dimensions and bitsizes.

  • 1000 dimensional matrices filled uniformly random with integers of 10, 100, or 1000 bits respectively.
  • matrices as they occur for the Knapsack problem with (dimension,bitsize) pairs of (10, 100000), (100,10000), (150,5000)
  • matrices as they appear for solving simultaneous Diophantine equations of (dim,bits) pairs (3, 128), (12, 10000), (76, 5000)
  • Ajtai (d, bits) (10, 7), (2, 13), (3, 11)
  • particular bad matrices with entries sized at 64, 128, and 500 bits.
  • NTRU (dim, bits, q) (10,100,12), (100,100,35), (5,1000,75)

days5/proj/lll (last edited 2008-11-14 13:41:57 by anonymous)