Prime Numbers

424 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
We have been intentionally brief in the final steps of the algorithm. The details
for these last stages are laid out splendidly in [Shor 1999]. The core idea
underlying the [Detect periodicity ...] step is this: After the FFT step, the
machine should be found in a final state | c 〉| x k mod n 〉 with probability
Pc,k =
1
q
q−1
a=0
x a ≡x k (mod n)
e 2πiac/q
2
⌊(q−k−1)/r⌋
=
1
e
q
b=0
2πi(br+k)c/q
2
. (8.4)
This expression, in turn, can be shown to exhibit “spikes” at certain rdependent
values of c. From these spikes—which we presume would all show
up simultaneously upon measurement of the QTM machine’s state—one can
infer after a quick side calculation the period r. See Exercises 8.22, 8.23, 8.24,
8.36 for some more of the relevant details. As mentioned in the latter exercise,
the discrete logarithm (DL) problem also admits of a QTM polynomial-time
solution.
Incidentally, quantum computers are not the only computational engines
that enjoy the status of being talked about but not yet having been built to
any practical specification. Recently, A. Shamir described a “Twinkle” device
to factor numbers [Shamir 1999]. The proposed device is a special-purpose
optoelectronic processor that would implement either the QS method or the
NFS method. Yet another road on which future computing machines could
conceivably travel is the “DNA computing” route, the idea being to exploit
the undeniable processing talent of the immensely complex living systems that
have evolved for eons [Paun et al. 1998]. If one wants to know not so much the
mathematical but the cultural issues tied up in futuristic computing, a typical
lay collection of pieces concerning DNA, molecular, and quantum computing
is the May-June 2000 issue of the MIT magazine Technology Review.
8.6 Curious, anecdotal, and interdisciplinary references
to primes
Just as practical applications of prime numbers have emerged in the
cryptographic, statistical, and other computational fields, there are likewise
applications in such disparate domains as engineering, physics, chemistry, and
biology. Even beyond that, there are amusing anecdotes that collectively signal
a certain awareness of primes in a more general, we might say lay, context.
Beyond the scientific connections, there are what may be called the “cultural”
connections. Being cognizant of the feasibility of filling an entire separate
volume with interdisciplinary examples, we elect to close this chapter with a
very brief mention of some exemplary instances of the various connections.
One of the pioneers of the interdisciplinary aspect is M. Schroeder, whose
writings over the last decade on many connections between engineering
and number theory continue to fascinate [Schroeder 1999]. Contained in
such work are interdisciplinary examples. To name just a few, fields Fq as