$[$Maximum mark: $7]$

$23$M$\mathrm{.}1.$SL$.$TZ

"Password entropy" is a measure of the predictability of a computer password. The higher the entropy, the more difficult it is to guess the password.

The relationship between the password entropy, $p,($measured in bits$)$ and the number of guesses, $G,$ required to decode the password is given by $0\mathrm{.}301p=lo{g}_{10}G.$

$($a$)$ Calculate the value of $p$ for a password that takes $5000$ guesses to decode.

$($b$)$ Write down $G$ as a function of $p.$

$($c$)$ Find the number of guesses required to decode a password that has an entropy of $28$ bits. Write your answer in the form $a{10}^k,$ where kinZ

There is a point on the graph of the function $G(p)$ with coordinates $(0,1).$

$($d$)$ Explain what these coordinate values mean in the context of computer passwords.