How is point addition for points of elliptic curve in $\mathbb{F}_p$ carried out technically?

in flag

From a very basic introduction text to elliptic curve cryptography point arithmetic is derived from "standard analysis": The (negative) sum of $P_1$ and $P_2$ is defined as the Point $P_3$, which is on the line connecting $P_1$ and $P_2$: enter image description here

From that it is derived

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In real numbers I would understand that completely. But typically, ECC is carried out within a finite (prime) field $\mathbb{F}_p$.

How shall I understand the multiplication/division in the addition formula $m \cdot (x_1-x_3) $ since m is a "fraction"? Normally I would expect that $m$ is a "fractional number" and in general not in $\mathbb{F}_p$. So how is $m$ obtained, because "division" is not an operation in a field - there is only an multiplicative inverse element:

Does it mean, I have to calculate

$y_3 = (y_2-y_1)(x_2-x_1)^{-1}(x_1-x_3) - y_1$

Since the inverse is well defined and exists, $y_3$ is also well defined. Is my assumption right or what else is the "fraction" $m$? Does it mean, the derivation gives sense, because formally each operation which is carried out in $\mathbb{R}$ I can in parallel be done also within $\mathbb{F}_p$, because both are fields and the notation $a/b$ means implicitly $a b^{-1}$?

MichaelW avatar
in flag
yes, this answers my question.
kelalaka avatar
in flag
Yes, it is common to write in that way since the formulas are generic for any field (C)..

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