Question

Perform encryption and decryption using the RSA algorithm, for the following p = 3, 9 = 11, e = 7 M = 5 What is the public key PU? What is the private key PR? What is the cipertext C? How does the decryption to covert the ciphertext back to the plaintext M?

Answer 1

Solution:

Given,

=>RSA algorithm is used

=>p = 3, q = 11, e = 7 and M = 5

Explanation:

Calculaitng value of n:

=>n = p*q

=>n = 3*11

=>n = 33

Calculating value of $\phi$ (n)(Euler totient of n):

=>$\phi$(n) = $\phi$ (p)*$\phi$(q)

=>$\phi$(n) = (p-1)*(q-1)

=>$\phi$(n) = 2*10

=>$\phi$(n) = 20

Calculating value of d(decryption constant):

=>We know that we need to choose d such that e*d mod$\phi$(n) = 1

=>7*d mod 20 = 1

=>d = 3 as 7*3 mod 20 = 1

Calculating value of PU(public key):

=>Pair (e, n) represents public key

=>Public key = (7, 33)

Calculating value of PR(private key):

=>Pair (d, n) represents value of private key.

=>Private key = (3, 33)

Calcualting value of ciphertext C:

=>We know that, ciphertext(C) = M^e mod n where M is plaintext

=>C = 5^7 mod 33

=>C = 14

Converting ciphertext to plaintext(M):

=>We know that, plaintext(M) = C^d mod n where C is ciphertext

=>M = C^d mod n

=>M = 14^3 mod 33

=>M = 5

I have explained each and every part with the help of statements attached to it.