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

In this case the remainder is 4, not 1. Why is this? The answer is that
15�3�5 and so is not a prime and Fermat’s Little Theorem only applies
when the modulus is a prime, such as 5, 7, 11 or 97, as in the examples
above. Euler showed how the theorem must be modified when the
modulus is not a prime. The original theorem though is
Fermat’s Little Theorem
If p is a prime number and m is any number which is not divisible by p then
m ( p�1) �1 (mod p),
i.e. m ( p�1) leaves remainder 1 when divided by p.
The proof is not difficult and generalises fairly easily to give a proof of the
Fermat–Euler theorem, which is given in M23.
The generalisation discovered and proved by Euler applies to any
modulus but the version required by the RSA system requires only the
case where the modulus is the product of just two distinct primes and so is
worth stating in its own right:
The Fermat–Euler Therorem (as needed in the RSA system)
If p and q are different prime numbers and m is any number which is not divisible by
p or q then
m ( p�1)(q�1) �1 (mod pq).
In the example above we had p�3, q�5 and m�2 and the theorem tells us
that
2 (2)(4) �1 (mod 15)
and indeed 2 8 �256�17�15�1.
Encipherment and decipherment keys in the RSA system
To encipher a text by means of the RSA method we require:
Encipherment and the internet 175
(1) a large number n (�pq) which is the product of just two distinct
primes, p and q (The question as to how one finds very large primes is
highly relevant. We have met this problem before, in connection with
the Diffie–Hellman key exchange system. In general a considerable
amount of computation is required. Since the primes to be used