To solve this problem, we need to decrypt the encrypted message chunks using the given public key. The decryption process involves finding the original message chunks

from the encrypted chunks

using the formula:

where

is the modular inverse of

modulo

$,$ and

is the Euler's totient function of

$.$ For

$,$ which is the product of two primes

and

$,$ we have:

a$)$ Finding the original message chunks

Calculate the modular inverse of

modulo

:

We need to find

such that:

Using the extended Euclidean algorithm, we solve for

:

$,$

Apply the Euclidean algorithm to find the greatest common divisor $($GCD$)$ and the coefficients:

$120=7*17+1$

$7=1*7+0$

From the first equation, we can express $1$ as:

Thus,

$($since

$).$

Decrypt each chunk:

Now, we use

to decrypt each chunk

:

For

:

For

:

For

:

For

:

These calculations can be efficiently performed using modular exponentiation.

b$)$ Computational challenges and modular exponentiation

Challenges:

Large Numbers: In real$-$time communication,

is typically very large $($hundreds or thousands of bits$),$ making direct computation of powers and modular reductions computationally expensive.

Efficiency: Calculating

directly for large

is inefficient due to the sheer size of the numbers involved.

Modular Exponentiation:

Optimization: Modular exponentiation allows us to compute

efficiently using the method of exponentiation by squaring. This reduces the number of multiplications required, making it feasible to handle large exponents.

Algorithm: The algorithm works by breaking down the exponentiation into a series of squarings and multiplications, reducing the problem size at each step.

By using modular exponentiation, secure messaging apps can perform encryption and decryption operations quickly, even with large keys, enabling real$-$time communication.