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

number of tests increased by a factor of only about 8. The explanation for
this can be found in M21.
This method of finding the prime factors of a number by dividing the
number by each prime less than its square root is, essentially, due to
Eratosthenes and is the standard method both for factorising (if it works)
and for showing that a number is a prime (if it fails). This is not the only
method that might be used, sometimes a short-cut can be found; for
example, someone might notice that
9167�9216�49�(96) 2 �(7) 2 �(96�7)(96�7)�89�103
and, even better, that
988 027�988 036 – 9�(994) 2 – (3) 2 �(994�3)(994�3)
�991�997
but, in general, we are not so lucky. Sometimes, such as when the number
that we are trying to factorise has a particular form such as
2 p –1
Encipherment and the internet 173
where p is a prime number, there are special techniques which reduce the
number of possibilities, but in the type of number which is relevant to the
RSA system these special techniques are not applicable.
The RSA system of encryption, which is described below, relies for its
security upon this fact: that it is very time-consuming to factorise a large
number even if we are told that it is the product of two large primes. As for
the encryption process in the RSA system the basis of this is an elegant and
powerful theorem stated, without proof, by the French mathematician
Pierre Fermat early in the seventeenth century. This is often referred to as
‘Fermat’s Little Theorem’ and is not to be confused with the notorious
‘Fermat’s Last Theorem’, which he also stated without proof, and which
was not proved until 1993 [13.2]. Fermat may have had a proof of his
‘Little Theorem’; it is extremely unlikely that he had a proof of his ‘Last
Theorem’. The Swiss mathematician Leonhard Euler gave a proof of
Fermat’s Little Theorem in 1760 and also generalised it, so giving us what
is known as the Fermat–Euler Theorem and it is this that is used in the
RSA encryption/decryption process.
As a preliminary it is instructive to look at some examples of