answer me on this one question what is the probability of hitting a random number that is in range 1B of the half of the target , even thought we are taking randoms until R>2^132 or R>2^133 .
Yeah sorry, you lost me. You can just divide by 1 billion to get the inverse of that probability.
Not sure what you mean by half point between target and whatever other point. That one cannot be computed, otherwise ECC would be broken.Because k can be either odd or even, so any halving may end up with a scalar on the other side of the curve.