# 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)