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Modern Cryptography and Elliptic Curves: A Beginner’s Guide

Thomas R. Shemanske

Chapter 6

A Little More Algebra - all with Video Answers

Educators

Chapter Questions

Problem 1

Exercise. Let
$$
\sigma=\left[\begin{array}{lll}
1 & 2 & 3 \\
2 & 3 & 1
\end{array}\right] \text { and } \tau=\left[\begin{array}{lll}
1 & 2 & 3 \\
1 & 3 & 2
\end{array}\right] \text {. }
$$

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Problem 2

Fill in the Cayley table for $S_3$ using the elements listed in the first row or column, and show that $S_3$ is non-abelian.
$$
\begin{array}{l||l|l|l|l|l|l}
\circ & e & \sigma & \sigma^2 & \tau & \sigma \tau & \sigma^2 \tau \\
\hline \hline e & & & & & & \\
\hline \sigma & & & & & & \\
\hline \sigma^2 & & & & & & \\
\hline \tau & & & & & & \\
\hline \sigma \tau & & & & & & \\
\hline \sigma^2 \tau & & & & & & \\
\hline
\end{array}
$$

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02:38

Problem 3

Compute $R, R^2, R^3, F, F^2, F^3, R F, R^2 F$.

Julie Silva
Julie Silva
Numerade Educator

Problem 4

Fill in the Cayley table for $D_3$ using the elements listed along the first row or column.
$$
\begin{array}{l||l|l|l|l|l|l}
\circ & e & R & R^2 & F & R F & R^2 F \\
\hline \hline e & & & & & & \\
\hline R & & & & & & \\
\hline R^2 & & & & & & \\
\hline F & & & & & & \\
\hline R F & & & & & & \\
\hline R^2 F & & & & & & \\
\hline
\end{array}
$$

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Problem 5

Notice that each symmetry can be thought of as a permutation of the three vertices. If we regard the numbers marking the vertices of the left-hand triangle as positions, then $R$ can be described as the permutation $R=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1\end{array}\right]$, and $F=\left[\begin{array}{lll}1 & 2 & 3 \\ 1 & 3 & 2\end{array}\right]$. Describe $R^2, F, R F, R^2 F$ in terms of the elements $\sigma$ and $\tau$ used to define $S_3$. Can you determine if $D_3 \cong S_3$ ?

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Problem 6

We know that $U_n$ is a finite abelian group. For $5 \leq n \leq$ 15, use your knowledge of these groups to characterize them as in the Fundamental Theorem. For example, $U_3$ is a group of order 2, a prime, so $U_3$ is a cyclic group of order 2 , that is $U_3 \cong \mathbb{Z}_2$. The group $U_8$ is an abelian group of order 4 , so by the Fundamental Theorem it is isomorphic to either $\mathbb{Z}_2 \times \mathbb{Z}_2$ or to $\mathbb{Z}_4$. We easily check for all $a \in U_8$ that $a^2=1$, so $U_8$ is not cyclic, and so $U_8 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$.

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