  The security of messages encoded via the widely used RSA public key
encryption system rests on the enormous computational effort required to find
the prime factors of a large number N using classical (i.e., conventional)
computers. In 1994, however, Peter Shor showed that for sufficiently large N a
quantum computer would be expected to perform the factoring with much less
computational effort. This paper endeavors to explain, in a fashion
comprehensible to the non-expert readers of this journal: (i) the RSA
encryption protocol; (ii) the various quantum computer manipulations
constituting the Shor algorithm; (iii) how the Shor algorithm performs the
factoring; and (iv) the precise sense in which a quantum computer employing
Shor's algorithm can be said to accomplish the factoring of very large numbers
with less computational effort than a classical computer can. It is made
apparent that factoring $N$ generally requires many successive runs of the
algorithm. The careful analysis herein reveals, however, that the probability
of achieving a successful factorization on a single run is about twice as large
as commonly quoted in the literature.
