Code and ciphers: Julius Caesar, the Enigma and the internet

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chapter 13
The opposite problem, finding two or more numbers which when
multiplied together produce a given number, is called factorisation. This is
a much harder task than multiplying numbers together, as anyone who
has tried it knows. For example: if we are asked to multiply 89 by 103 we
should be able to get the answer, 9167, in less than a minute. If, however,
we were asked to find two numbers whose product is 9167 it would probably
take us considerably longer. How would we do it?
The standard method of factorisation
If we are asked to factorise a large number, N, we should use the fact that
unless N is a prime number it must have at least two factors and the
smaller of these cannot exceed the square root of N. This means that in
the case where N�9167 we need only test for divisibility by the primes
less than �9167, which is nearly 96. The largest prime below 96 is 89, so
in this case we would succeed on the very last test, by which time we
would have carried out more than 20 divisions. Had we carried out the
tests on N�9161 we would have failed to find a divisor, since 9161 is a
prime.
As N increases so does the number of tests that we have to make. Thus
if N�988 027 N either is a prime or is divisible by some number less than
�988 027, which is a little less than 994. We would then try dividing
988 027 by each prime less than 994. If we find a prime that divides
988 027 exactly, i.e. without leaving a remainder, we have solved the
problem. If no such number is found we would know that 988 027 is
prime. In fact
988 027�991�997
and since both 991 and 997 are prime numbers the factorisation is complete.
It would have taken us quite a lot of effort to do this because there
are more than 160 primes less than 991, and we would have had to try
them all before we were successful. Even with a calculator this would be a
time-consuming and tedious job. Someone who has a computer and can
program could, of course, get the computer to do the work. Irrespective of
how it was done, by increasing N from 9167 to 988 027, a factor of about
108, we, or the computer, were faced with an increase in the number of
divisions from about 20 to over 160. Note that although N increased by a
factor of over 100, so that �N increased by a factor of more than 10, the