Nigel Smart's attack solves the discrete logarithm problem in linear time. It requires the curve, however, to be anomalous, i.e. to have a trace of Frobenius equal to one or, equivalently, to be of the same order as the underlying field, $\#E=p$.
Smart's paper is here: http://www.hpl.hp.com/techreports/97/HPL-97-128.html
I'm trying to understand why it doesn't work for other curves? I'm guessing that it has something to do with the properties of the p-adic logarithm but I can't quite put my finger on it.
Any hint or explanation will be appreciated.