hi all i read the post since one year sorry for my english i found a interesting think but i does takes me further. in every elliptic curve like y^2= x^3+7 there is something interesting like :
if P(1,y1) - k times--> Q(-29/3 ,y2)
P(2,y3) --k times--> Q(-3,y2)
so on there is a simple math here where k is always too know independent from whcih curve we work .
i don't want to give more information this operation is 10 times faster then k*G= and find the x value
Too bad the points need to be on the curve. We already know about the endomorphisms.
I'd say any attempt to break a private key that involves more than a constant amount of scalar multiplications (no matter how well optimized by precomputed tables), has very few chances of success.
Any multiplication means, by definition, more than one addition, more time.
Random key -> multiply and match -> good luck waiting.
First level of magnitude reduction: don't do scalar multiplications.
Second wall to break is then the point addition (and there's one more after that, and finally one more after). I already said too much, but I believe there's something that can run around 20x in less time (fewer computations) if we know the public key and simplify the question. Some known details around secp256k1 help a lot.