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any possible classical computer for certain classes of problems that are considered
intractable for classical computers.
1.1.5 Shor’s Algorithm
Perhaps the most practical example of this is the problem of factoring the
product N of two large primes. The best currently known classical algorithm, the
general number field sieve [Lenstra and Lenstra, 1993], requires sub-exponential
time O(e (log N)1/3 (log log N) 2/3 ), while a quantum computing algorithm proposed by
Peter Shor in 1997 requires polynomial time O((log N) 3 ) [Shor, 1997].
The algorithm is based on reducing the factoring problem to the problem of
finding the period r of the function f(x) = a x mod N, where a is a random number
less than N. If a and r fulfill certain requirements, one of the factors of N is the
gcd(a r/2 ± 1, N), otherwise the procedure is repeated for a new random value for
a. This approach makes use of the quantum computer’s power by employing it for
the period finding step. Since the quantum computer can evaluate f(x) for many
values of x simultaneously, it can very quickly generate the period r by fourier
transforming interference patterns of different evaluations of f(x).
Many of the modern data encryption schemes, e.g. RSA, are based on the
intractability of the factoring problem. Therefore there is great interest by intelligence
agencies to develop a system that can implement Shor’s algorithm to gauge
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