The mploc software package
The mploc software package
Canonical lift
Using mploc, is is currently relatively easy to compute for instance
the canonical lift (Teichmüller lift) of a given definition polynomial up to arbitrary precision.
Let
be a finite field. Consider an
unramified extension
of
. A given
element
can be lifted to many different
elements. The canonical lift of
is the
unique element of
which maps to
in the
residue field and which is at the same time a root of the
polynomial
in
.
When the input data is an irreducible polynomial
over
, consider an extension
of
, and an unramified extension
of
having
as a residue field. The
canonical lift of
is the minimal polynomial of the
canonical lift of
in
, where
is a root of
in
. The
canonical lift is defined over
.
The mploc distribution contains code in the demo/
directory, more precisely files teich1.c and
teich2.c (see also the main calling code in
teich.c), which compute the canonical lift of a given
irreducible polynomial over
using various
algorithms.
At present, the code in teich2.c in mploc-0.3
achieves the following
timings for lifting a polynomial of degree 163 up to the given precision.
Timings are in seconds on a 3.00GHz Intel Pentium 4 processor.
| precision | time |
| 256 | 0.04 |
| 512 | 0.08 |
| 1024 | 0.17 |
| 2048 | 0.36 |
| 4096 | 0.78 |
| 8192 | 1.73 |
| 16384 | 3.95 |
| 32768 | 7.72 |
| 65536 | 22.33 |