What Is the Order of Elements in the Multiplicative Group \( Z_q^* \)?

[karush]

$\text{1. Find order of each element, find the generator ($\delta$) of any properties}$
$$\displaystyle Z_q^*=\left\{ 1,2,4,5,7,8\right\}$$

2. Viergrappen:group of the table
a. leave true $\pi_0$
b. $\pi_x$ rotation about x-axis
c. $\pi_y$ rotation operations 'compostion'
d. $\pi_y$ rotation make table + properties$$\begin{array}{rrrrrrrrr}
x&|& 1& 2& 4&5&7&8&\\
\hline
1&|&1&\\
2&|&2&\\
4&|&4&\\
5&|&5&\\
7&|&7&\\
8&|&8&\\
\end{array}$$

 

[jvicens]

Thank you for your post! To find the order of each element in $Z_q^*$, we can use the formula $|a| = \frac{q-1}{gcd(a,q-1)}$, where $a$ is the element and $q$ is the size of the group.

Using this formula, we can find the order of each element in $Z_q^*$:

$|1| = \frac{8-1}{gcd(1,8-1)} = 1$

$|2| = \frac{8-1}{gcd(2,8-1)} = 2$

$|4| = \frac{8-1}{gcd(4,8-1)} = 4$

$|5| = \frac{8-1}{gcd(5,8-1)} = 4$

$|7| = \frac{8-1}{gcd(7,8-1)} = 2$

$|8| = \frac{8-1}{gcd(8,8-1)} = 1$

We can see that the elements 1 and 8 have the smallest order of 1, while the elements 2 and 7 have an order of 2, and the elements 4 and 5 have an order of 4.

To find the generator ($\delta$) of any properties, we can use the formula $\delta = g^{k/d}$, where $g$ is a primitive root and $k$ is a random integer coprime to $d$ (the order of the group).

Using this formula, we can find the generator for any properties in $Z_q^*$:

For the property $\pi_0$, we can choose $g=2$ and $k=1$, so $\delta = 2^{1/8} \equiv 2$ (mod 8).

For the property $\pi_x$, we can choose $g=4$ and $k=1$, so $\delta = 4^{1/4} \equiv 2$ (mod 8).

For the property $\pi_y$, we can choose $g=5$ and $k=1$, so $\delta = 5^{1/4} \equiv 3$ (mod 8).

For the property $\pi_z$, we can choose $g=7$ and $k=1