Alice received the following ciphertext from Bob, $"081408".$ Bob had encrypted it using the RSA cypher with Alice's public key

$($pq$,$ e$)=(55,3),$

where

p $=5$

and

q $=11.$

Note that

$($p $1)($q $1)=40.$

The value for d in Alice's private key,

$($pq$,$ d$)$

is a positive inverse for

$3$ modulo $($p $1)($q $1).$

It was found to be $27$ in Example $8\mathrm{.}4\mathrm{.}8($b$)$ and Example $8\mathrm{.}4\mathrm{.}10.$ What is Bob's message after Alice decrypts it$?($Assume Bob encoded one letter at a time using the encoding

A $=01,$ B $=02,$ C $=03,,$ Z $=26.)$

To decrypt Bob's message, Alice uses the decryption formula

$$

M $=$ C mod $,$

where M is the code for a letter of the message, C is the encrypted version of the letter,

$($pq$,$ e$)=(55,3)$

is the public key, and

$($pq$,$ d$)=(55,27)$

is the private key.

$($a$)$

To begin, Alice computes the values of

a$,$ b$,$ c$,$ d and e

that are indicated below.

$081$a $($mod $55)082$ b $($mod $55)084$ c $($mod $55)$

$088$d $($mod $55)0816$ e $($mod $55)$

She finds that

a $=,$

b $=,$

c $=,$

d $=,$

and

e $=.$

Because

$27=16+8+2+1,0827=0816+8+2+1=0816088082081,$

she uses the values of

a$,$ b$,$ d$,$ and e

to compute

$0827$ mod $55=($a $$ b $$ d $$ e$)$ mod $55=.$

Thus, the first letter in Bob's message is $.$

$($b$)$

Alice finds the second letter of Bob's message by computing

$27$ mod $55=.$

$($c$)$

What is Bob's message after Alice finishes decrypting it$?$