Point of elliptic curve

Question

How can we calculate the multiple of a point of an elliptic curve?

For example having the elliptic curve $y^2=x^3+x^2-25x+39$ over $\mathbb{Q}$ and the point $P=(21, 96)$.

To find the point $6P$ is the only way to calculate:

1. the point $2P=P+P$,
2. then $4P=2P+2P$
3. and then finally the point $6P=4P+2P$ ?

Or is there also an other way of calculation?

EDIT:

$$P=(21, 96)$$

$$2P=P+P=\left ( \frac{13153}{2304}, \frac{1185553}{110592} \right )$$

$$4P= \left (-\frac{21456882568875649}{3238354750023936} , \frac{3395969291284125120479041}{122855718046564076691456} \right )$$

$$6P=\left (\frac{26455920935919644458805579323004114785}{14704264997379508491439452468204834816} , -\frac{1075150031960164636335160890473952630299280887362209417804659119}{66847620865553399763849555951358904102466015610213125405278208} \right )$$

Can someone check if the coordinates of the point is correctly calculated?

Answer 1

Sure there are other ways. For example,

$2P=P+P$, $3P=2P+P$, $6P=3P+3P$

Also,

$2P=P+P$, $3P=2P+P$, $4P=3P+P$, $5P=4P+P$, $6P=5P+P$

and so on.

In general, the double and add method you describe is the fastest. It is akin to the square and multiply that you often see in multiplicative groups.

Answer 2

Sage gives the following $2P=\left(\frac{13153}{2304} , \frac{1185553}{110592}\right),$ $6P=\big(\frac{17631797546863867480163645661711294049}{2834578067615933833996300908324147456} ,-\frac{60902529607177336000181399672827762453069546262535228527}{4772353810493036247904139120367622993558177805319376896} \big{)}$ $=(6.220254699738563,-12.761528592718786)$