on algorithm, to be summarized and illustrated in the next section. Shor's factorization algorithm The rewards of going through the preceding sections and tedious developments are now at hand. In fact, this section does not bring any new concept in this respect. It only consists in the formalization of Shor's factorization algorithm. There are many possible ways to formalize an algorithm, from a list of practical steps with programming flow charts, to more mathematically abstract and academic definitions. Here, I shall follow the first approach and also provide some illustrations. It is brought to the reader's attention that date, there exists no classical computer algorithm making it possible to implement order- finding in polynomial time. The order-finding algorithm, the key constituent in Shor's factorization, is to be implemented in a hypothetical quantum-computing or quantum- phase-estimation circuit, as illustrated in Fig. 20.3. As we have seen, the rest is only a matter of classical computation with an overall O(L³) or O[(log N)³ complexity. We may first make a few assumptions to simplify the algorithm description. First, the composite number N must be odd. The test is immediate, based on the value of the last or lowest-weight binary digit. Second, it must not be a trivial product of small prime numbers and their powers. For instance N = 15, 27, 39, 144 are trivial composites for the human brain. For a supercomputer, which has in its memory the list of the first 1000 prime numbers (for instance), and look-up tables of the type shown in Table 20.2 giving the basic factorizations, thereof, there exist millions of "trivial" composites that can be factorized in milliseconds or faster. In such cases, there is no point whatsoever in considering Shor's algorithm! Another trivial case (as viewed from a computer perspective) is when there exist two integers a > 1 and b > 2, such that N = ab. It is an academic issue, not addressed here, to establish that the solution a, b can be classically obtained in O(L³) time. T