Encrypt and decrypt by means of the RSA algorithm with the following system parameters:1. p = 3, q = 11, d = 7, x = 52. p = 5, q = 11, e = 3, x = 9

Sure, let's go step by step for each of the two scenarios:

1. p = 3, q = 11, d = 7, x = 5

Step 1: Compute n = pq = 311 = 33

Step 2: Compute φ(n) = (p-1)(q-1) = 210 = 20

Step 3: Choose e such that 1 < e < φ(n) and e and φ(n) are coprime. Let's choose e = 3.

Step 4: Ensure that the 'd' value satisfies the equation (de) mod φ(n) = 1. For d = 7, the equation is (73) mod 20 = 1, which is true.

Step 5: Public key is (e, n) => (3, 33) and private key is (d, n) => (7, 33)

Step 6: To encrypt x = 5, we calculate y = (x^e) mod n = (5^3) mod 33 = 125 mod 33 = 26

Step 7: To decrypt y = 26, we calculate x = (y^d) mod n = (26^7) mod 33 = 5

2. p = 5, q = 11, e = 3, x = 9

Step 1: Compute n = pq = 511 = 55

Step 2: Compute φ(n) = (p-1)(q-1) = 410 = 40

Step 3: Ensure that the 'e' value satisfies the condition 1 < e < φ(n) and e and φ(n) are coprime. For e = 3, the conditions are satisfied.

Step 4: Compute 'd' such that (d*e) mod φ(n) = 1. The value of 'd' that satisfies this equation is 27.

Step 5: Public key is (e, n) => (3, 55) and private key is (d, n) => (27, 55)

Step 6: To encrypt x = 9, we calculate y = (x^e) mod n = (9^3) mod 55 = 729 mod 55 = 14

Step 7: To decrypt y = 14, we calculate x = (y^d) mod n = (14^27) mod 55 = 9

So, the encryption of x = 5 with the first set of parameters is y = 26 and the decryption of y = 26 is x = 5. The encryption of x = 9 with the second set of parameters is y = 14 and the decryption of y = 14 is x = 9.