# 21.7 Computer Problems

Let $E$ be the elliptic curve ${y}^{2}\equiv {x}^{3}+2x+3\text{\hspace{0.17em}}(\mathrm{m}\mathrm{o}\mathrm{d}\text{\hspace{0.17em}}19)\text{.}$

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

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

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

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

Show that $(1\text{,}\text{}5)$ has exactly 20 distinct multiples, including $\text{\u221e}\text{.}$

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 $(x\text{,}\text{}y)$ on the curve ${y}^{2}\equiv {x}^{3}+7x+11\text{\hspace{0.17em}}(\mathrm{m}\mathrm{o}\mathrm{d}\text{\hspace{0.17em}}593899)\text{.}$ 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 $x\text{-coordinate}$.

Factor 3900353 using elliptic curves.

Try to factor 3900353 using the $p-1$ method