Solution

Concept: 

RSA Algorithm:

Step 1: Calculate value of n = p × q, where p and q are prime no.’s

Step 2: calculate Ø(n) = (p-1) × (q-1)

Step 3: consider d as a private key such that Ø(n) and d have no common factors. i.e  greatest common divisor (Ø(n) , d ) = 1

Step 4: consider e as a public key such that (e × d) mod Ø(n) = 1.

Step 5: Ciphertext = message i.e. me mod n.

Step 6: message= cipher text i.e. cd mod n.

Calculation:

To find the value of 'd' in the RSA algorithm, we need to calculate the modular multiplicative inverse of 'e' modulo φ(n), where n is the product of the two prime numbers p and q, and φ(n) is the Euler's totient function.

Given:
p = 13
q = 5
e = 7
ciphertext = 6

First, calculate n:
n = p × q
n = 13 × 5
n = 65

Next, calculate φ(n):
Ø(n) = (p - 1) × (q - 1)
Ø(n) = (13 - 1) × (5 - 1)
Ø(n)= 12 × 4
Ø(n) = 48

Now, we need to find the modular multiplicative inverse of 'e' modulo Ø(n). In other words, we need to find 'd' such that (e * d) mod Ø(n) = 1.

Using the extended Euclidean algorithm, we can find 'd':

Step 1: 
48 = 7 × 6 + 6
7 = 6 × 1 + 1

Step 2:
6 = 1 * 6 + 0

Since the remainder in the last step is 1, we can conclude that the greatest common divisor of 7 and 48 is 1. Therefore, 'd' exists, and it is the coefficient of 7 in the equation:

1 = 7 - 6 × 1

So, 'd' is equal to 7.

Now, to decrypt the ciphertext using the private key (d, n), we can use the formula:

plaintext = (ciphertext^d) mod n

Substituting the values:
plaintext = (6⁷) mod 65

Calculating this:

plaintext = 279936 % 65
plaintext = 46

Therefore, the value of 'd' is 7 and the plaintext (decrypted value) of the ciphertext '6' using the private key (d, n) is 46.

So, the correct answer is 7, 46.