In Sage 8.0 I find the following unexpected behavior:

sage: k.<a,b> = GF(2)[]
sage: K.<x> = k[]
sage: f = x^1000 + a*x^500 + b
sage: time f^16
CPU times: user 7.61 s, sys: 8 ms, total: 7.62 s
Wall time: 7.62 s
x^16000 + a^16*x^8000 + b^16

Is it really the case that the exponentiation does not make use of the 
positive characteristic? Of course I could write a quick function which 
does what I need in *reasonable* time. But somehow I hope that Sage 
provides the most basic arithmetic operations, and I simply missed it. What 
puzzles me even more is this: f^16 takes 7.61 s, but f^1024 takes only 2.04 
s. While the latter time is still ridiculous, it seems that for larger 
exponents some better (yet very poor) algorithm is used.

-- Peter Mueller

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