More than a dozen jokes - thanks guys and girls!
(plus some anti-French abuse from the usual suspect).
Well my brain doesn't hurt so much any more, and it was well worth it.
I've now got an even faster method for elliptic-curve point counting,
It lifts a curve over a field of degree n in time O(n^(2+1/2+eps)),
or O(n^(2+eps)) with precomputation.  Before the best methods took
O(n^(3+eps)) without precomputation, or O(n^(2+1/2+eps)) with it.
The precomputation is done once per field, not per curve, and takes
time O(n^(3+eps)).  Here eps is an arbitrarily small number, hiding
After lifting, you compute a norm in time O(n^(2+1/3+eps)) to get the
Here's an example over a 1009-bit field, without precomputation, using
INFO: -=-=-=-=-= ECPC: Elliptic Curve Point Counting, made easy! =-=-=-=-=-
INFO: Picked field polynomial 1+x^55+x^1009.
INFO: Checking... OK OK OK OK OK OK OK OK OK OK
/   \           .-.      Software Development       .-.           /   \

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"Around the world, there is a growing sense that democracy has not
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