Addition of Points

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

Example 1:

We consider the curve E:y2=x3+x+1E: y^{2}=x^{3}+x+1 over F5\mathbb{F}_{5} of this example :Example_4_5_3. Let P=(x1,y1)=(2,4)P=\left(x_{1}, y_{1}\right)=(2,4) and Q=(x2,y2)=(4,3)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 aβˆ’1=apβˆ’2=a5βˆ’2=a3a^{-1}=a^{p-2}=a^{5-2}=a^{3} and hence

2βˆ’1=23=8=3modΒ 52^{-1}=2^{3}=8=3 \quad mod \space5

and

4βˆ’1=43=64≑4modΒ 5,4^{-1}=4^{3}=64 \equiv 4 \quad mod \space 5,

respectively.

It’s x1β‰ x2x_{1} \neq x_{2}, hence, by applying the addition formulas (1) of this proposition :Explicit_formulas_for_addition , we calculate

x3=(y2βˆ’y1x2βˆ’x1)2βˆ’x1βˆ’x2=(y2βˆ’y1)2((x2βˆ’x1)2)βˆ’1βˆ’x1βˆ’x2=(3βˆ’4)2((4βˆ’2)2)βˆ’1βˆ’2βˆ’4=(βˆ’1)2(22)βˆ’1βˆ’2βˆ’4=1β‹…4βˆ’1βˆ’2βˆ’4=1β‹…43βˆ’2βˆ’4=1β‹…64βˆ’2βˆ’4=58≑3modΒ 5\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

y3=(y2βˆ’y1x2βˆ’x1)(x1βˆ’x3)βˆ’y1=(3βˆ’4)(4βˆ’2)βˆ’1(2βˆ’3)βˆ’4=(βˆ’1)(2)βˆ’1(βˆ’1)βˆ’4=(βˆ’1)2(2)3βˆ’4=1β‹…8βˆ’4=8βˆ’4=4.\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)P+Q=(2,4)+(4,3)=(3,4).

Example 2

Table shows the addition table for the elliptic curve EE : y2=x3+1y^{2}=x^{3}+1 over F5\mathbb{F}_{5} of this example :Example_4_5_2.

Example 3

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

Table 1

Addition table for the curve E:y2=E: y^{2}= x3+1x^{3}+1 over F5\mathbb{F}_{5}.

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