Allan Steel (October 2007)
Magma V2.14 (released in October 2007) has many improvements for computations with finite fields and polynomials over finite fields. The following tables show a comparison with the previous release V2.13.
All timings are for an 2.8GHz Opteron processor running Fedora Core 6 Linux. I will expand this as I get time.
We factor the polynomial xn + x + 1 over GF(2) for various values of n.
We factor the polynomial xn + x + w over K=GF(3k) for various values of k and n, where w is the default generator of K. We use w as the constant to force computations over K proper, since xn + x + 1 is comparatively trivial to factor over such fields. Prime values of k are chosen, since these are the hardest extension fields to work with (there is no subfield to be taken advantage of).
We multiply two random polynomials of increasing degree n over GF(21000). Since V2.14 uses the Cantor algorithm, the timings for it do not grow smoothly.