Elliptic curves 2P, 3P

Question

How do I compute 2P, 3P etc?

ex: $y^2=x^3+4xmod7$

and I have to compute the order of (2,3)=P

and my example says 2P =(0,0) 3P=(2,4) but I don't know how to get these answers?

Answer 1

You should let a computer try all possibilities for nP, until nP=O. Then n* is the order of P.

EDIT: To do this using a computer or by hand, use the formulae found on wikipedia: http://en.m.wikipedia.org/wiki/Elliptic_curve_point_multiplication

Instead of division, you should multiply by the modular multiplicative inverse, since we are working in a finite field. The modular multiplicative inverse can be found by using the extended Euclidean algorithm.