EllipticCurve

Elliptic Curves $E(\mathbb{R})$ over the Field of Real Number $\mathbb{R}$ Defined by Weierstrass Equation $E: y^2=x^3+ax+b$ where $a,b\in\mathbb{R}$ and $4a^3+27b^2\neq 0$

Definition. Elliptic curve $E(\mathbb{R})$ over real number $\mathbb{R}$ defined by Weierstrass equation $E: y^2=x^3+ax+b$, where $a,b\in\mathbb{R}$ and $\Delta=4a^3+27b^2\neq 0$, is the set of all solutions of the equation together with a special point called point at infinity denoted by $\mathcal{O}.$ We write $$E(\mathbb{R})=\{(x,y)\in\mathbb{R}\times\mathbb{R}\text{ such that }y^2=x^3+ax+b\}\cup\{\mathcal{O}\}$$

On $E(\mathbb{R})$ we define point addition as the following rules:

• Negation of the point at infinity $\mathcal{O}$ denoted by $-\mathcal{O}$ is $\mathcal{O}$ itself. We write $-\mathcal{O}=\mathcal{O}$.
• Negation of a point $P=(x,y)$ is second point of intersection points between vertical line through $P$ and the curve $E(\mathbb{R})$. In this case $-P=(x,-y)$.
• $\mathcal{O}+\mathcal{O}=\mathcal{O}$ and $P+(-P)=\mathcal{O}.$
• If $P_1\neq\pm P_2$, then $P_1+P_2$ is the reflection point of the third intersection point between the elliptic curve and the line $(P_1P_2).$
• If $P_1=P_2$, then $P_1+P_2=2P_1$ is the reflection point of the third intersection point between the elliptic curve and the tangent line at $P_1.$

Remark. The point at infinity $\mathcal{O}$ is sitting on top, or sleeping at the bottom of $y$-axis. To connect a point $P$ to $\mathcal{O}$ we draw vertical line through $P$. The point at infinity $\mathcal{O}$ has no affine coordinate representation.

1. Let $E(\mathbb{R})$ be an elliptic curve over real number defined by equation $E: y^2=x^3+1$.
   1. Find the discriminant of the equation.
   2. Show that $P_1(-1,0)$ and $P_2(0,1)$ are on the curve.
   3. Write the equation of line $(L)$ through $P_1$ and $P_2$.
   4. Find the third intersection point $Q(x_Q,y_Q)$ between $(L)$ and the curve $E(\mathbb{R})$.
   5. Find the reflection point $P_3(x_3,y_3)$ of $Q$ across the $x$-axis.
   6. Find the implicit derivative of the equation $E$.
   7. Write the equation of tangent line $(T)$ at $P_2$.
   8. Find the third intersection point $R(x_R,y_R)$ between $(T)$ and the curve $E(\mathbb{R})$.
   9. Find the reflection point $P_4(x_4,y_4)$ of $R$ across the $x$-axis.

2. Let $E(\mathbb{R})$ be an elliptic curve over real number defined by equation $E: y^2=x^3-x$.
   1. Find the discriminant of the equation.
   2. Show that $P_1(-1,0)$ and $P_2(0,0)$ are on the curve.
   3. Write the equation of line $(L)$ through $P_1$ and $P_2$.
   4. Find the third intersection point $Q(x_Q,y_Q)$ between $(L)$ and the curve $E(\mathbb{R})$.
   5. Find the reflection point $P_3(x_3,y_3)$ of $Q$ across the $x$-axis.
   6. Find the implicit derivative of the equation $E$.
   7. Write the equation of tangent line $(T)$ at $P_2$.
   8. Find the third intersection point $R(x_R,y_R)$ between $(T)$ and the curve $E(\mathbb{R})$.
   9. Find the reflection point $P_4(x_4,y_4)$ of $R$ across the $x$-axis.

3. Let $E(\mathbb{R})$ be an elliptic curve over real number defined by equation $E: y^2=x^3-2x+1$.
   1. Find the discriminant of the equation.
   2. Show that $P_1(0,1)$ and $P_2(1,0)$ are on the curve.
   3. Write the equation of line $(L)$ through $P_1$ and $P_2$.
   4. Find the third intersection point $Q(x_Q,y_Q)$ between $(L)$ and the curve $E(\mathbb{R})$.
   5. Find the reflection point $P_3(x_3,y_3)$ of $Q$ across the $x$-axis.
   6. Find the implicit derivative of the equation $E$.
   7. Write the equation of tangent line $(T)$ at $P_2$.
   8. Find the third intersection point $R(x_R,y_R)$ between $(T)$ and the curve $E(\mathbb{R})$.
   9. Find the reflection point $P_4(x_4,y_4)$ of $R$ across the $x$-axis.

4. Let $E(\mathbb{R})$ be an elliptic curve over real number defined by equation $E: y^2=x^3-4x+1$.
   1. Find the discriminant of the equation.
   2. Show that $P_1(0,-1)$ and $P_2(0,1)$ are on the curve.
   3. Write the equation of line $(L)$ through $P_1$ and $P_2$.
   4. Find the third intersection point $Q(x_Q,y_Q)$ between $(L)$ and the curve $E(\mathbb{R})$.
   5. Find the reflection point $P_3(x_3,y_3)$ of $Q$ across the $x$-axis.
   6. Find the implicit derivative of the equation $E$.
   7. Write the equation of tangent line $(T)$ at $P_2$.
   8. Find the third intersection point $R(x_R,y_R)$ between $(T)$ and the curve $E(\mathbb{R})$.
   9. Find the reflection point $P_4(x_4,y_4)$ of $R$ across the $x$-axis.

5. Let $E(\mathbb{R})$ be an elliptic curve over real number defined by equation $E: y^2=x^3-4x+4$.
   1. Find the discriminant of the equation.
   2. Show that $P_1(-2,2)$ and $P_2(2,-2)$ are on the curve.
   3. Write the equation of line $(L)$ through $P_1$ and $P_2$.
   4. Find the third intersection point $Q(x_Q,y_Q)$ between $(L)$ and the curve $E(\mathbb{R})$.
   5. Find the reflection point $P_3(x_3,y_3)$ of $Q$ across the $x$-axis.
   6. Find the implicit derivative of the equation $E$.
   7. Write the equation of tangent line $(T)$ at $P_2$.
   8. Find the third intersection point $R(x_R,y_R)$ between $(T)$ and the curve $E(\mathbb{R})$.
   9. Find the reflection point $P_4(x_4,y_4)$ of $R$ across the $x$-axis.

6. Let $E(\mathbb{R})$ be an elliptic curve over real number defined by equation $E: y^2=x^3-6x+5$.
   1. Find the discriminant of the equation.
   2. Show that $P_1(-2,3)$ and $P_2(2,-1)$ are on the curve.
   3. Write the equation of line $(L)$ through $P_1$ and $P_2$.
   4. Find the third intersection point $Q(x_Q,y_Q)$ between $(L)$ and the curve $E(\mathbb{R})$.
   5. Find the reflection point $P_3(x_3,y_3)$ of $Q$ across the $x$-axis.
   6. Find the implicit derivative of the equation $E$.
   7. Write the equation of tangent line $(T)$ at $P_2$.
   8. Find the third intersection point $R(x_R,y_R)$ between $(T)$ and the curve $E(\mathbb{R})$.
   9. Find the reflection point $P_4(x_4,y_4)$ of $R$ across the $x$-axis.

7. Let $E(\mathbb{R})$ be an elliptic curve over real number defined by equation $E: y^2=x^3+ax+b$ with $4a^3+27b^2\neq 0$. Let $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ be two non-zero (not point at infinity) distinct points on the curve.
   1. Give formula for computing negation point $-P_1(x_3,y_3)$ of $P_1(x_1,y_1)$.
   2. Find the implicit derivative of the equation $E$.
   3. Write formula for computing slope of a tangent line at $P_1(x_1,x_2)$.
   4. Show that $P_1=-P_2$ if and only if $x_1=x_2$ or $y_1=-y_2$.
   5. Write addition formula of $P_1+P_2=P_3$ if $P_1\neq -P_2$.
   6. Show that tangent at $P_1$ is vertical if and only if $y_1=0$.
   7. Write doubling formula of $2P_1=P_3$ if $y_1\neq 0$.

Theorem. Let $E(\mathbb{R})$ be an elliptic curve over real number $\mathbb{R}$ defined by Weierstrass equation $E: y^2=x^3+ax+b$ where $a,b\in\mathbb{R}$. Let $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ be two non-zero (not the point at infinity) points on the curve and one is not the negation of another. Then the addition formula for $P_1+P_2=P_3$ is given by $$\begin{cases}x_3=\lambda^2-x_1-x_2\\ y_3=\lambda(x_1-x_3)-y_1\end{cases}\;\text{where}\;\lambda=\begin{cases}\frac{y_2-y_1}{x_2-x_1}&\text{if}\;P_1\neq P_2\\ \frac{3x_1^2+a}{2y_1}&\text{if}\;P_1=P_2\end{cases}.$$

Remark. $\lambda$ is either the slope of line through $P_1$ and $P_2$ or the slope of tangent line at $P_1$.

Theorem. Elliptic curve $(E(\mathbb{R}),+)$ forms a commutative group with identity element $\mathcal{O}$.

Elliptic Curve over Prime Field

Consider elliptic curve over prime field $GF(p)$, $p>3$ is a prime, defined by Weierstrass equation $E: y^2=x^3+ax+b$ where $a,b\in GF(p)$. Since $GF(p)$ is of order $p$, the order of $E(GF(p))$ is then less than or equal to $2p+1$. The elliptic curve $E(GF(p))$ is finite.

1. Let $E(GF(7))$ be an elliptic curve over $GF(7)$ defined by $E: y^2=x^3+2x+4$.
   1. Compute the discriminant of the equation.
   2. Verify that $P_1(0,2)$ and $P_2(1,0)$ are on the curve.
   3. Compute the addition point $P_3=P_1+P_2$ and doubling point $P_4=2P_1$.
   4. List of all points on the curve.

Application of Elliptic Curves in Internet Browsers