Learn the order of a point and the group order in this lesson.

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Example 1:

We consider the curve $E: y^{2}=x^{3}+x+1$ over $\mathbb{F}_{5}$ of this example :Example_4_5_3. Let $P=\left(x_{1}, y_{1}\right)=(2,4)$ and $Q=\left(x_{2}, y_{2}\right)=(4,3)$. We precompute the multiplicative inverses of 2 and 4. By this proposition :proposition_4_6_2

, it holds that $a^{-1}=a^{p-2}=a^{5-2}=a^{3}$ and hence

$2^{-1}=2^{3}=8=3 \quad mod \space5$

and

$4^{-1}=4^{3}=64 \equiv 4 \quad mod \space 5,$

respectively.

It’s $x_{1} \neq x_{2}$, hence, by applying the addition formulas (1) of this proposition :Explicit_formulas_for_addition , we calculate

\begin{aligned} x_{3} &=\left(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\right)^{2}-x_{1}-x_{2}=\left(y_{2}-y_{1}\right)^{2}\left(\left(x_{2}-x_{1}\right)^{2}\right)^{-1}-x_{1}-x_{2} \\ &=(3-4)^{2}\left((4-2)^{2}\right)^{-1}-2-4=(-1)^{2}\left(2^{2}\right)^{-1}-2-4=1 \cdot 4^{-1}-2-4 \\ &=1 \cdot 4^{3}-2-4=1 \cdot 64-2-4=58 \equiv 3 \quad mod \space 5 \end{aligned}

and

\begin{aligned} y_{3} &=\left(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\right)\left(x_{1}-x_{3}\right)-y_{1} \\ &=(3-4)(4-2)^{-1}(2-3)-4=(-1)(2)^{-1}(-1)-4=(-1)^{2}(2)^{3}-4 \\ &=1 \cdot 8-4=8-4=4. \end{aligned}

Thus, we conclude that $P+Q=(2,4)+(4,3)=(3,4)$.

Example 2

Table shows the addition table for the elliptic curve $E$ : $y^{2}=x^{3}+1$ over $\mathbb{F}_{5}$ of this example :Example_4_5_2.

Example 3

Consider the two points $(0,1)$ and $(0,4)$ of Example 2. Since both have the same $x$-coordinate, it holds that $(0,1)+(0,4)=\mathcal{O}$ as we can easily check from Table 1.

Table 1

Addition table for the curve $E: y^{2}=$ $x^{3}+1$ over $\mathbb{F}_{5}$.

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