Let $E$ be the elliptic curve $y^2\equiv x^3+2x+3(\mathrm{m}\mathrm{o}\mathrm{d}19)\text{.}$

1. Find the sum $(1\text{,}5)+(9\text{,}3)\text{.}$

2. Find the sum $(9\text{,}3)+(9\text{,}-3)\text{.}$

3. Using the result of part (b), find the difference $(1\text{,}5)-(9\text{,}3)\text{.}$

4. Find an integer $k$ such that $k(1\text{,}5)=(9\text{,}3)\text{.}$

5. Show that $(1\text{,}5)$ has exactly 20 distinct multiples, including $\text{∞}\text{.}$

6. Using (e) and Exercise 19(d), show that the number of points on $E$ is a multiple of 20. Use Hasse’s theorem to show that $E$ has exactly 20 points.

You want to represent the message $12345$ as a point $(x\text{,}y)$ on the curve $y^2\equiv x^3+7x+11(\mathrm{m}\mathrm{o}\mathrm{d}593899)\text{.}$ Write $x=12345\mathrm{\_}$ and find a value of the missing last digit of $x$ such that there is a point on the curve with this $x\text{-coordinate}$.

1. Factor 3900353 using elliptic curves.

2. Try to factor 3900353 using the $p-1$ method of Section 9.4. Using the knowledge of the prime factors obtained from part (a), explain why the $p-1$ method does not work well for this problem.

Let $P=(2\text{,}3)$ be a point on the elliptic curve $y^2\equiv x^3-10x+21(\mathrm{m}\mathrm{o}\mathrm{d}557)\text{.}$

1. Show that $189P=\text{∞}\text{,}$ but $63P\ne \text{∞}$ and $27P\ne \text{∞}\text{.}$

2. Use Exercise 20 to show that $P$ has order 189.

3. Use Exercise 19(d) and Hasse’s theorem to show that the elliptic curve has 567 points.

Compute the difference $(5\text{,}9)-(1\text{,}1)$ on the elliptic curve $y^2\equiv x^3-11x+11(\mathrm{m}\mathrm{o}\mathrm{d}593899)\text{.}$ Note that the answer involves large integers, even though the original points have small coordinates.