The best algorithm known for finding logarithms on an elliptic curve (E) is the (parallelized) Pollard lambda collision search. We show how to apply a Pollard lambda search on a set of equivalence classes derived from E, which requires fewer iterations than the standard approach. In the case of anomalous binary curves over F2m , the new approach speeds up the standard algorithm by a factor of 2m.
Robert P. Gallant, Robert J. Lambert, Scott A. Van