Shor Algorithm

1
Shor's Algorithm for Quantum Factorization
Procedure Pick a pseudo-random number a lt N
Compute gcd(a, N). This may be done using the
Euclidean algorithm. If gcd(a, N) ? 1, then it
is a nontrivial factor of N, so we are done.
Otherwise, use the period-finding subroutine
(below) to find r, the period of the following
function f(x) a x (mod N) If r is odd, go
back to step 1. If a r/2 -1 (mod N), go back
to step 1. The factors of N are gcd(ar/2 1,
N). We are done.
2
Greatest Common Divisor
The greatest common divisor (abbreviated GCD), or
highest common factor (HCF) of two integers is
the largest integer that divides both numbers.
The GCD of a and b is often written as
gcd(a,b). For example, gcd(12,18) 6 and
gcd(5,0) 5. The GCD of 0 and 0 is usually
defined to be 0. Two numbers are called coprime
or relatively prime if their greatest common
divisor equals 1. For example, 9 and 28 are
relatively prime. While the GCD of two numbers
can in principle be computed by determining the
prime factorizations of the two numbers and
comparing factors, this is never done in
practice, because it is too slow. A much more
efficient method is the Euclidean algorithm.
3
Euclidean Algorithm

• The Euclidean algorithm is an extremely fast
  algorithm to determine the greatest common
  divisor (GCD) of two integers.
• The algorithm does not require factoring.
• Given two non-negative integers a and b, first
  check if b is zero. If yes, then the GCD is a. If
  no, calculate c, the remainder after the division
  of a by b. Replace a with b, b with c, and start
  the process again.
• For example, the GCD of 1029 and 42 is computed
  by this algorithm to be 21 with the following
  steps
• __a_____b___
• 42
• 42 21
• 21 0

4
Modular Arithmetic
Modular arithmetic is a modified system of
arithmetic for integers, sometimes referred to as
'clock arithmetic', where numbers 'wrap around'
after they reach a certain value (the modulus).
For example, while 8 6 equals 14 in
conventional arithmetic, in modulo 12 arithmetic
the answer is two, as two is the remainder after
dividing 14 by the modulus 12. If a is any
integer and n is a positive integer, we write a
mod n for the remainder in 0, ..., n-1 that
occurs if a is divided by n. For instance, 26 mod
12 2. Similarly, 7642956 mod 107 53. We
call two integers a, b 'congruent modulo n' iff
their difference is divisible by n, or
equivalently if they leave the same remainder
when divided by n, i.e. if a mod n b mod n. In
this case, we write a b (mod n) For instance
14 26 (mod 12)
5
RSA
RSA is an asymmetric algorithm for public key
cryptography. Credit for its development is
given to Ron Rivest, Adi Shamir and Len Adleman,
who described the algorithm in 1977, the letters
RSA being the initials of their surnames. The
security of the RSA system relies on the
difficulty of factoring very large numbers. RSA
is widely used in electronic commerce. The
algorithm was patented by MIT in 1983 in the
United States of America. The patent expired in
September 2000. Since the algorithm had been
published prior to the patent application, it
could not be patented in other countries. Most
authorities suggest that 1024 bit keys will be
secure for some time, barring a fundamental
breakthrough in factoring practice, but others
favor even longer keys.