A secure messaging app uses modular arithmetic to encrypt user messages. Suppose a

message is divided into numeric chunks, and each chunk m is encrypted using the formula c $=$

$($m$^$e mod n$),$ where e and n are public encryption keys. For a particular user, the public key is

e$=7$ and n$=143.$ The encrypted message is received as the sequence: $[18,36,121,30].$

a$)$ Find the original message chunks by calculating the modular inverse using the extended

Euclidean algorithm.

b$)$ Explain the computational challenges involved in real$-$time communication when using

large values of n and how modular exponentiation helps optimize this process.