Introduction to Cryptography with Coding Theory, 3rd Edition

21.6 Exercises

1. 1. Let $x^3+a{x}^2+bx+c$ be a cubic polynomial with roots $r_1\text{,}{r}_2\text{,}{r}_3\text{.}$ Show that $r_1+r_2+r_3=-a\text{.}$

   2. Write $x=x_1-a/3\text{.}$ Show that

      $$x^3+a{x}^2+bx+c=x_1^3+b^{\prime }{x}_1+c^{\prime }\text{,}$$

      with $b^{\prime }=b-(1/3)a^2$ and $c^{\prime }=c-(1/3)ab+(2/27)a^3\text{.}$ (Remark: This shows that a simple change of variables allows us to consider the case where the coefficient of $x^2$ is 0.)

2. Let $E$ be the elliptic curve $y^2\equiv x^3-x+4(\mathrm{m}\mathrm{o}\mathrm{d}5)\text{.}$

   1. List the points on $E$ (don’t forget $\text{∞}$).

   2. Evaluate the elliptic curve addition $(2\text{,}0)+(4\text{,}3)\text{.}$

3. 1. List the points on the elliptic curve $E:y^2\equiv x^3-2(\mathrm{m}\mathrm{o}\mathrm{d}7)\text{.}$

   2. Find the sum $(3\text{,}2)+(5\text{,}5)$ on $E\text{.}$

   3. Find the sum $(3\text{,}2)+(3\text{,}2)$ on $E\text{.}$

4. Let $E$ be the elliptic curve $y^2\equiv x^3+x+2(\mathrm{m}\mathrm{o}\mathrm{d}13)\text{.}$

   1. Evaluate $(1\text{,}2)+(2\text{,}5)\text{.}$

   2. Evaluate $2(1\text{,}2)\text{.}$

   3. Evaluate $(1\text{,}2)+\text{∞}\text{.}$

5. 1. Find the sum of the points (1, 2) and (6,3) on the ...