The second theorem establishes that the probability of the period meeting certain eligibility criteria is at least 75% for any composite. These two theorems combined validate and conclude Shor's factorization algorithm. The factorization of the composite N = 15 = 3 X 5 is found in textbooks as the only illustrative example of Shor's algorithm. Here, we shall investigate the whole space of nontrivial composites N < 100, as an emulation of the quantum computer. It is possible to do this based on the fact that for such relatively small numbers, we can compute (with a basic home computer) all the possibilities associated with each step of Shor's algorithm, namely what the period-finding quantum circuit should yield, and the associated probabilities of success or failure. The result of this investigation is an original plot showing the probability II of successfully concluding the factorization of nontrivial composites N < 100 in a single run. The exercise helps one to grasp how Shor's algorithm would work when taking greater composite numbers. The last section, which briefly describes public key cryptography (PKC), is not completely out of place in this chapter. The purpose of this addition is to show how the PKC algorithm works, as based on the product N = pq of two prime numbers p, q, whose factorization is indeed considered intractable by classical means. Should a quantum computer of corresponding computing power be implemented someday, the whole field of PKC-based cryptography, and Internet security for that matter, would be compromised overnight! Fortunately, this remains a distant perspective, while from this chapter, we know that the theory works mathematically. Phase estimation This section considers an algorithm referred to as phase estimation. The word "esti- mation" is correct, because the outcome of the algorithm is only what can be called a good estimate of the phase in a quantum system. This phase estimation will enable us to move another step further in our progression towards Shor's factorization algo