Please implement the RSA encryption and decryption algorithm from scratch. RSA is an asymmetric cryptographic algorithm that uses two keys$$a public key for encryption and a private key for decryption. Your implementation should be able to:

$1.$ Compute a public key $($encryption key$)$ and the private key $($decryption key$)$ using the Extended Euclidean Algorithm.

$2.$ Encrypt a message using the public key.

$3.$ Decrypt the message using the private key.

Here are some key requirements for your implementation. Please READ CAREFULLY:

$1.$ Prime Numbers:

$$ Use two small prime numbers for p and q$,$ both smaller than $100.$ You can directly assign the two prime numbers, so no prime generation is needed in your code.

$$ You can use the values p $=61$ and q $=83$ in your code.

$2.$ Public Key:

$$ You can use e$=17$ as the public exponent.

$3.$ Key Calculation:

$$ Compute the private key d using the Extended Euclidean Algorithm $($EEA$).$

$4.$ Fast Exponentiation:

$$ Implement the square$-$and$-$multiply algorithm to efficiently compute modular exponentiation for both encryption and decryption processes.

$5.$ Encryption and Decryption:

$$ Use the formula c $=$ me mod$$n for encryption, where mmm is the plaintext message.

$$ For decryption, use m $=$ cd mod$$n to retrieve the original message.

$6.$ Test Case:

$$ Provide an example test case with a numerical message m$=65$ to verify both encryption and decryption. Here is the test case in Python. You can adjust it to any other language.

# Example usage

p $=61$

q $=83$

# Public key generation

public$\_$key, private$\_$key $=$ generate$\_$keys$($p$,$ q$)$

$$

# Encrypting a message

message $=65$ # Example plaintext message

ciphertext $=$ encrypt$($message$,$ public$\_$key$)$

print$($f$"$Encrypted message: $\{$ciphertext$\}")$

# Decrypting the message

$$decrypted$\_$message $=$ decrypt$($ciphertext$,$ private$\_$key$)$

print$($f$"$Decrypted message: $\{$decrypted$\_$message$\}")$

$$

Report Requirements:

$$ Your report should document the procedures and implementation of RSA, including key generation, encryption, decryption, and fast exponentiation. Please show your understanding of the RSA algorithm