Introduction to Cryptography with Coding Theory, 3rd Edition

22.7 Exercises

1. Let $E$ be the supersingular elliptic curve from Section 22.1.

   1. Let $P\ne \infty$ and $Q\ne \infty$ be multiples of $P_0\text{.}$ Show that $\tilde{e}(P\text{,}Q)\ne 1\text{.}$ (Hint: Use the fact that $\tilde{e}(P_0\text{,}{P}_0)=\omega$ is a $q$th root of unity and that ${\omega }^x=1$ if and only if $x\equiv 0(\text{mod}q)\text{.}$)

   2. Let $Q\ne \infty$ be a multiple of $P_0$ and let $P_1\text{,}{P}_2$ be multiples of $P_0\text{.}$ Show that if $\tilde{e}(P_1\text{,}Q)=\tilde{e}(P_2\text{,}Q)\text{,}$ then $P_1=P_2\text{.}$

2. Let $E$ be the supersingular elliptic curve from Section 22.1.

   1. Show that

      $$\tilde{e}(aP\text{,}bQ)=\tilde{e}(P\text{,}Q{)}^{ab}$$

      for all points $P\text{,}Q$ that are multiples of $P_0\text{.}$

   2. Show that

      $$\tilde{e}(P+Q\text{,}R)=\tilde{e}(P\text{,}R)\tilde{e}(Q\text{,}R)$$

      for all $P\text{,}Q\text{,}R$ that are multiples of $P_0\text{.}$

3. Let $E$ be the supersingular elliptic curve from Section 22.1. Suppose you have points $A\text{,}B$ on $E$ that are multiples of $P_0$ and are not equal to $\infty \text{.}$ Let $a$ and $b$ be two secret integers. ...