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 through this step by step.

1. For the first system parameters:

   • First, we need to calculate n and φ(n). n is the product of p and q, and φ(n) is the product of (p-1) and (q-1). So, n = pq = 311 = 33 and φ(n) = (p-1)(q-1) = 210 = 20.

   • The encryption key e is calculated as the multiplicative inverse of d mod φ(n). But in this case, d is given as 7. So, e = d mod φ(n) = 7 mod 20 = 7.

   • Now, we can encrypt x. The encryption formula is c = x^e mod n. So, c = 5^7 mod 33 = 25.

   • To decrypt c, we use the formula x = c^d mod n. So, x = 25^7 mod 33 = 5. This is the original message.

2. For the second system parameters:

   • Again, we calculate n and φ(n). So, n = pq = 511 = 55 and φ(n) = (p-1)(q-1) = 410 = 40.

   • The encryption key e is given as 3.

   • We encrypt x using the formula c = x^e mod n. So, c = 9^3 mod 55 = 49.

   • To find d, we need to calculate the multiplicative inverse of e mod φ(n). So, d = e^-1 mod φ(n) = 3^-1 mod 40 = 27.

   • Now, we can decrypt c using the formula x = c^d mod n. So, x = 49^27 mod 55 = 9. This is the original message.