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If a generator point G is used to create a prime point and this generated point is then divided by any integer, the result gives a non-integer multiple of G.

Example: 2G / 4 = 0.5G

Does this point lie on the curve?

If it does not lie on the curve, what would be the most appropriate way to represent a point such as 0.5G on the elliptic curve graph?

Edit: Assume secp256k1 parameters.

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    It depends on which kind of field the elliptic curve is defined. If finite field is used, there're only finite number of points in the curve and if the divisor is coprime with the number of points, we could transform the division into mulitplication. In other cases, the result could be not available or there could be multiple results – Zhaohui Du Sep 04 '19 at 01:45
  • Interesting. Assume we are using the secp256k1. I updated the question. –  Sep 04 '19 at 01:53

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The curve is defined in a finite field $F_p$ and the number of points (order) in the ECC group is a big prime n. Cofactor of the ECC group to be 1 shows that all points in the curve could be generated by G.

To find X/m where m is coprime with n, we could first find the integer k so that $mk=1(\mod n)$ (Extend Euclidean algorithm could be used). After that we could replace X/m by kX.

Zhaohui Du
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  • I see. Do you know what would be the appropriate integer scalar to use on a point such as 10.5G? Example: point 10.5G can be represented as 1727459G. –  Sep 05 '19 at 14:18
  • What is your meaning when you use "appropriate"? For any integer m between 1 and n-1, the multiplication mG is fine. – Zhaohui Du Sep 06 '19 at 07:02
  • As far as I know all points can be reached with integers between 0 and n - 1. With appropriate I mean integer value. –  Sep 06 '19 at 09:23