Points

EPPDISCMATH$58\mathrm{.}4\mathrm{.}020.$

Bob wants to send Alice the message, "KEY," which he plans to encrypt using Alice's RSA cypher with public key $(pq,e)=(55,3).$ To encrypt the message, Bob uses the method described in Example $8\mathrm{.}4\mathrm{.}9.$ He encodes one letter at a time using $A=01,B=02,C=03,$dots,$Z=26.$ Next, he applies the encrypting formula

$C=M^e$modpq,

where $M$ is a plaintext letter and $C$ is a block of ciphertext. Because $55$ is a two$-$digit integer, each block of ciphertext is a two$-$digit integer with $0,1,2$dots,$9$ represented as $01,02,03,$dots,$09.($Enter your answers using a fixed number of digits: $01$ for $1,02$ for $2,$dots,$09$ for $9,10,11,$ etc.$)$

$($a$)$ When the first letter of the message is encoded, the result is

$?3$mod$55=.$ Bob then applies the encrypting formula to find that the first two$-$digit block of the encrypted message is $$

$$

$($b$)$ The second two$-$digit block of Bob's encrypted message is $q,{?}^3$mod$55=.$

Enter an exact number.

$($c$)$ What is the fully encrypted message that Alice receives? $($Enter the message as a sequence, of integer doubles separated by a single space, where each double is written using a fixed number of digits: $01$ for $1,02$ for $2,$dots,$09$ for $9,10,11,$ etc.$)$

$$

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