I'd like to implement point multiplication on elliptic curves over prime fields. My problem is that I've found different definition how to do it.
At adding: the second parameter of the result is not tha same on WIKI and on WolframMathWorld. Why are the different? I think it's simple but I don't get it.
And a side-question: what is the simplest way to check my results? I've found a few calculator applet, but they give different results, so I can't say which one is right.
The Wikipedia entry is correct. The sum of two points is supposed to be the reflection across the $x$-axis of the third point of intersection of $E$ with the line through $P$ and $Q$. The equation of the line through $P$ and $Q$ is $y = \lambda (x - x_1) + y_1$. So we need to plug in $x_3$ and then take the negative, giving $$y_3 = \lambda(x_1 - x_3) - y_1.$$