How to get a common coordinate from two different coordinates on Elliptic Curves?

Dew Debra

2/19/23, 11:12 PM

I am trying to write a SageMath script that multiplies two coordinates on Elliptic Curves into one common coordinate.

SageMath Elliptic curves over finite fields documentation

p = Number

M1 = EllipticCurve (GF (p), [0,7])

C1 = M1 ([x1, y1])
C2 = M1 ([x2, y2])
C3 = C1 * C2

Somewhere they wrote that using sets of the SageMath function it is possible to do this. How to do it?

0 + 0

elliptic-curves

keys

discrete-logarithm

arithmetic

elliptic-curve-generation

fgrieu

2/20/23, 5:09 AM

Before computing something, one must have a clear picture of what it is. We can add two points on an Elliptic Curve. We can multiply a point on an Elliptic curve by an integer (that's mathematically defined by repeated addition, and well-defined since addition is associative). But there is no common definition of the multiplication of two points on an Elliptic Curve, unless we also mention a generator $G$, in which case we can define that $C_1=c_1⋅G$ and $C_2=c_2⋅G$ implies by definition that $C_1*C_2=(c_1×c_2)⋅G$ where $×$ is modular multiplication modulo the order $n$ of $G$.

Daniel S

2/20/23, 9:38 AM

If you're trying to find an implementation of the elliptic curve group law in sagemath, it uses the "+" operator. In other words, `C3=C1+C2` will set `C3` equal to the combination of `C1` and `C2` under the elliptic curve group law.

Dew Debra

2/20/23, 9:43 AM

@DanielS How do I write `" + "operator` and this whole formula in `SageMath`?

kelalaka

2/20/23, 10:04 AM

For addition: `C3 = C1 + C2` this works since you defined C1 and C1 as points. For multiplication: Does this answer your question? [How do I multiply two points on an elliptic curve?](https://crypto.stackexchange.com/questions/88214/how-do-i-multiply-two-points-on-an-elliptic-curve), If not, please indicate your actual aim.

Maarten Bodewes

2/20/23, 1:33 PM

You can always [edit] your document and comment below to get it reopened, assuming of course that there is then enough information to distinguish it from the other Q/A.

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: How to get a common coordinate from two different coordinates on Elliptic Curves?

TH: จะรับพิกัดร่วมจากสองพิกัดที่แตกต่างกันบน Elliptic Curves ได้อย่างไร

RO: Cum să obțineți o coordonată comună din două coordonate diferite pe curbele eliptice?

RU: Как получить общую координату из двух разных координат на эллиптических кривых?

VI: Làm cách nào để có tọa độ chung từ hai tọa độ khác nhau trên Đường cong Elliptic?

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