Hello I am struggling with q. 10 I have answered all the rest and they provide the needed information to solve 10. The questions that i had to anwser are next to the question number.

1) Let the prime numbers be 3 and 7

2) Calculate n = p * q

n = 3 * 7

n = 21

3) Calculate z = (p - 1) (q-1).

z = (3-1) (7-1)

z = 2 * 6

z = 12

4) Chose a prime number k so that k is co-prime to z. This just means that the greatest common divisor between k and z (12) must be 1 so I chose 5.

gcd(12,5) = 1

k = 5

5) Take note of n and k.

n = 12 , k =5

6)Now, use the following equation to calculate.

5 * j = 1 (mod 12)

5j - rz = 1

5j - 12r = 1

j = 1 + 12r / 5

We will apply trial and error to figure out r to find the smallest positive integer that satisfies the congruence.

j = 1 + 12(2) / 5 = 24 + 1 /5 = 25/5 = 5

j = 25 is the smallest positiev integer that satisfied the congruence.

7) Take note of j

j = 25

8) Choose any number P

P = 4

9) Calculate. Take note of.

4^5 ( mod 12 ) = E

E = 1024 (mod 12)

E = 4

10)Now, assume you forgot what your number P was. The way to get that number P back is using this equation:.Calculate,using your valuesfor.

4^25 = P ( mod 12 )

= P (mod 12)