Indeed -- they were using some nontrivial algebra in Sage. Note that
they write SAGE, and their bibliography reference is
William Stein et al. ... 2009.
Hopefully their version of Sage isn't really that old! In the article
they mention installing "a SAGE server version 1.4". Can it be?!
Also of note, they mention on page 18,19 (at the pagebreak) a
calculation that was not supported "at the time of writing" -- finding
an inverse to a certain element in a quotient of a polynomial ring, I
think. They do some algebra by hand to simplify the question to one
which their version of Sage can solve, but I wonder if someone who is
familiar with these things can check whether the current version of
Sage really doesn't support their calculation.
-Niles