# Integers in ECC

Let A be a point on curve with integral coordinates. Does k.A necessarily have integer coordinates? If so than why and if not than how to find A and k such that k.A has integral coordinates.

• I have seen that, what I do not understand is that why are the final coordinates necessarily integral. – nikhil_vyas May 28 '14 at 17:51
• why is slope integral? – nikhil_vyas May 28 '14 at 17:54
• You mean, why is $(y_2 - y_1)/(x_2 - x_1)$ integral? Well, if we're doing an elliptic curve on a finite field, then the above operation is within the field -- that means that, assuming $x_2 \ne x_1$, that the result of the above equation is a field element. – poncho May 28 '14 at 18:19

If you look at elliptic curves over $\mathbb C$, and do point addition with points with integer coordinates (as $\mathbb Z \subset \mathbb C$), then the result of the point addition usually will not have integer coordinates.
But in cryptography, we don't use elliptic curves over $\mathbb C$, but over a finite field $\mathbb F$. So the coordinates are not integers, but field elements. Subtracting and dividing field elements by each other (assuming no division by zero) gives you new field elements.