Network security and encryption is also a concern of a network administrator. Many encryption schemes are based on number theory and prime numbers; for example, RSA encryption. These methods rely on the difficulty of computing and testing large prime numbers. (A prime number is a number that has no divisor except for itself and 1.)

For example, in RSA encryption, one must choose two prime numbers, p and q; these numbers are private but their product, z = pq, is public. For this scheme to work, it is important that one cannot easily find p or q given z, which is why p and q are generally large numbers.

Choose an example of p and q and compute their product z. Justify your selection.

Assume that you wish to make a risk assessment and you wish to determine how probable it may be for a hacker to determine p and q from z. You wish to use discrete probability for this computation. For the sake of example, you choose to assess z = 502,560,410,469,881. Say that a hacker will attempt to find p and thus q by randomly selecting a potential divisor and testing to see if it divides 502,560,410,469,881. (You know that p = 15,485,867 and q = 32,452,843, but the hacker does not.) For example, the hacker may choose 1021; however, upon inspection the hacker will see that 1021 does not divide z.

For all questions below, please show all your work and/or justify your answers.

Given this problem, what is the sample space of the problem? Hint: In this context, the sample space is the set of all possible values that the hacker may select.

Given the sample space defined above, what events correspond to a successful guess by the hacker? Hint: An event is a subset of the sample space.

Given the above, what is the probability that the hacker will successfully guess p and/or q?

Assume the hacker selects five numbers to test.

What is the probability that all five attempts will fail? Show your work.

What is the probability that one of the five attempts will succeed? Show your work.