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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.

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Yes. You should review the basics of ECC point multiplication. – fgrieu May 28 '14 at 17:47
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
up vote 3 down vote accepted

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.

If you represent field elements by integers, then the new coordinates are "integers" again.

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