$2\mathrm{.}3.$ Encryption Related Questions

Question $5$: Public$-$key Cryptography$-$ Miscellaneous questions

$[18$ points as shown$]$

It is not considered safe to use RSA with small prime numbers. Suppose a corporation uses a pair of small prime numbers to generate their public and private keys.

As an attacker, you can access the public key $($e and n $)$ and an encrypted message, c$,$ as shown below.

Public key $($e$,$ n$)$: e $=11,$ n $=91$

Encrypted message: c $=29$

Task $1$: $[10$points$]$ Can you find the private key $(\backslash (\backslash$mathrm$\{$d$\},\backslash$mathrm$\{$n$\}\backslash )),$ by finding the value of d $?$ Use that to find the original message, $\backslash ($ m $\backslash ).$

Task $2$: $[4$ points$]$ Is it possible to find the original message $\backslash ($ m $\backslash )$ without finding the private key d by some brute$-$forcing technique? Explain the difficulty of using this approach in attacking some real$-$world application that uses RSA.

Task $3$: $[4$ points$]$ For the same value of $\backslash ($ n $\backslash )$ as above, can we use $3$ as the value of e$?$ Why yes or why not? Please elaborate on your answer

End of Question $5$: Public$-$key Cryptography$-$ Miscellaneous questions