# What is the point at infinity on secp256k1 and how to calculate it?

I hear that there should be a point at infinity on secp256k1. I wounder how to calculate it and what does it even mean. I tried to calculate it as $$P_{inf}=P+(-P)$$ but this gives different results for different P. Are there more than one point at infinity?

Geometrically thinking on an elliptic curve, adding two points that are symmetric against x-axis, should not be feasible because any vertical line on x-y plane will only intersect the curve at TWO points not THREE.

# Point at Infinity

Yes, there is a point at infinity $$O$$ (or denoted as $$\infty$$) on Elliptic Curves (EC) as the identity element of the Elliptic curve group. Let the EC be given in Weierstrass equation;

$$y^2 = x^3 +a x + b$$

$$O$$, the point at infinity can have different values according to the underlying coordinate system;

# Coordinate Systems

For an elliptic curve over $$\mathbb{F}_p$$;

• In Homogeneous Coordinates (also known as Projective Coordinate), the points are defined as $$(x:y:z),z\neq 0$$ corresponding to affine point $$(\frac{x}{z},\frac{y}{z})$$. The point at infinity is $$(0:1:0)$$ and negative value of $$(x:y:z)$$ is $$(x:-y:z)$$
• In Jacobian Coordinate the points are defined as $$(x:y:z),z\neq 0$$ corresponding to affine point $$(\frac{x}{z^2},\frac{y}{z^3})$$. The point at infinity is $$(1:1:0)$$ and negative value of $$(x:y:z)$$ is $$(x:-y:z)$$

Now, what is $$O$$ in affine coordinates? Try to convert $$O$$ back into an affine coordinate! Well, we cannot represent it! See this answer Thus, we simply say it is $$O$$. But, magically, it can be encoded in implementations.

# The Group Law

The points on an elliptic curve form an additive group with an identity $$O$$. For geometrical meaning please see the below image from Wikipedia Elliptic Curves The group law.

## The Group Law on Affine Coordinates

Arithmetically can define the addition rules in affine coordinates. These are derived from the line intersection and tangent equations. The formulas are;

Let $$P=(x_1,y_1)$$ and $$Q=(x_2,y_2)$$ be two point in the elliptic curve.

1. $$P+O=O+P=P$$
2. If $$x_1 = x_2$$ and $$y_1 = - y_2$$, that is $$Q =(x_2,y_2)=(x_1,−y_1)=−P$$, then $$P+Q = P + (-P) = O$$
3. If $$Q \neq -P$$ then the addition $$P+Q = (x_3,y_3)$$ and the coordinate can be calculated by;

\begin{align} x_3 = & \lambda^2 -x_1 - x_2 \mod p\\ y_3 = & \lambda(x_1-x_3) -y_1 \mod p \end{align}

$$\lambda = \begin{cases} \frac{y_2-y_1}{x_2-x_1}, & \text{if P \neq Q} \\[2ex] \frac{3 x_1^2+a}{2y_1}, & \text{if P = Q} \\[2ex] \end{cases}$$

## The Group Law on projective Coordinates

Let $$P_i = (x_i : y_i : z_i ), i = 1, 2$$, be points on the elliptic curve $$y^2 z = x^3 + Axz^2 + Bz^3.$$ Then $$(x_1 : y_1 : z_1 ) + (x_2 : y_2 : z_2 ) = (x_3 : y_3 : z_3 ).$$ The formulas are;

1. $$P_1 \neq \pm P_2$$
• $$u = y_2 z_1 − y_1 z_2,$$
• $$v = x_2 z_1 − x_1 z_2,$$
• $$w = u^2 z_1 z_2 − v^3 − 2v^2 x_1 z_2,$$ then

$$x_3 = vw, \quad y_3 = u(v^2 x_1 z_2 − w) − v^3 y_1z_2, \quad z_3 = v^3 z_1 z_2$$

1. $$P_1 = P_2$$
• $$t = a z_1^2 + 3x_1^2,$$
• $$u = y_1 z_1,$$
• $$v = u x_1 y_1,$$
• $$w = t^2 − 8 v,$$

$$x_3 = 2uw, \quad y_3 = t(4v − w) − 8y_1^2 u^2 , \quad z_3 = 8 u^3.$$

For Jacobian Coordinate operations see this link.

• In 2 you mean P + Q = O (or P + -P) Jan 6, 2019 at 10:13
• @kelalaka the Wikipedia link you have supplied is saying that if P and Q are opposites of each other, we define $P + Q = O$. Why we cannot conclude $P+(-P)=O$ is always true? By the way what is the random looking output value is that my computer program in python is giving as output? finally is it possible to calculate $O$ so that for any randomly given $P$ the computer program can calculate $P+O=P$? Jan 6, 2019 at 10:57
• The points are forming an additive group with identity $O$ so $P+(-P) = O$. Wikipedia also says the opposite of a point $P$ is $-P$. For the programming part, unfortunately, off-topic here. Please ask a question on StackOverflow. Jan 6, 2019 at 11:17

What does (the point at infinity) even mean?

That's an abstract point/concept required so that the addition of points on the Elliptic Curve is a group law, including

• Addition of any two elements $$P$$ and $$Q$$ of the group is an element of the group
• There is a neutral element $$\mathcal O$$ such that for all $$P$$, $$P+\mathcal O=P=\mathcal O+P$$.

A possible definition: the point at infinity is a single extra element $$\mathcal O$$ added to the curve and the rules of point addition, such that for all $$P$$ on the curve (including the point at infinity $$\mathcal O$$), it holds $$P+\mathcal O=P=\mathcal O+P$$.

We can then define $$-P$$ as the point $$Q$$ such that $$P+Q=\mathcal O=Q+P$$, and for all $$P$$ it holds $$P+(-P)=\mathcal O=(-P)+P$$.

How to calculate (the point at infinity)

One does not, much like one does not calculate $$+\infty$$.

One may define a conventional representation of the point at infinity in the Cartesian coordinate system by picking any coordinate not on the curve, e.g. $$(x,y)=(0,0)$$ assuming $$b\ne0$$. With this convention, a point $$(x,y)$$ is on the curve if $$(x,y)=(0,0)$$ or if $$y^2 = x^3 +a x + b$$. And the addition law is extended as

$$(x_0,y_0)+(x_1,y_1)=\begin{cases} (0,0)&\text{ if }(x_0,y_0)=(x_1,-y_1)\\ (x_0,y_0)&\text{ if }(x_1,y_1)=(0,0)\\ (x_1,y_1)&\text{ if }(x_0,y_0)=(0,0)\\ \text{per usual rules }&\text{ otherwise} \end{cases}$$

As noted in this comment, the point at infinity has a more natural representation in projective coordinates.

A correct implementation of point addition should give a representation of the point at infinity $$\mathcal O$$ when asked to compute $$P+(-P)$$ for $$P$$ any point on the curve. Note that in Cartesian coordinates, if $$P=(x,y)$$ and $$P\ne\mathcal O$$, then $$-P=(x,-y)$$ or $$-P=(x,(p-y)\bmod p)$$ depending on conventions. The representation of $$\mathcal O$$ is not necessarily unique, therefore $$P+(-P)=Q+(-Q)$$ does not necessarily hold.

• I like that angle of connecting with the group law that requires neutral element of the group operation. Another nice connection would be through projective coordinates - for projective coordinates (x,y,z) we have an affine representation (x/z. y/z) and the only point that cannot be represented that way is the case when z=0 which is exactly the point at infinity.
– Kris
Feb 18, 2020 at 13:10

For the secp256k1 curve, there is a point of infinity which is N the total points for this curve. To calculate use scalar multiplication of N and the base generator point for the curve. Another way is to go 2coin.org, go to the private key hex tab, enter the number 1 for the private key press enter. Then subtract by pressing -1 you will be at infinity, press one more time, you are at N-1.

• That result does not satisfy the curve equation. It means that it is not on the curve! Feb 24, 2020 at 16:06