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

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chapter 12
one or more keys which only they know. The problem is ‘How are they to
let each other know their secret key(s) without other people discovering
them too?’
The key distribution problem
The situation is that X and Y wish to communicate with each other using
an agreed encryption system. A third party, Z, knows their agreed encryption
system, is able to intercept their messages and would like to be able
to read them. X and Y may or may not know of the existence of Z but they
want to be sure that their messages should be unintelligible to anyone
other than themselves. The system, which they must assume is known to
Z or to anyone else, requires the use of one or more keys, which they
intend to keep secret but which may be changed from time to time, possibly
with each message, or possibly less frequently. Anyone who gets
hold of the keys and knows the method of encryption can decipher their
messages. It is therefore essential that the keys remain secret, but how can
X and Y tell each other their keys without running the risk that Z will
intercept and be able to exploit them?
The Diffie–Hellman key exchange system
An elegant solution to the key exchange problem was proposed by Diffie
and Hellman in 1976 [12.6]. Their method is implemented by X and Y as
follows.
(1) X and Y agree upon the use of two integers p and m (say) where p is a
large prime and m lies between 1 and ( p�1). The values of p and m
need not be kept secret.
(2) X chooses a secret number, x, and Y chooses a secret number, y. Both x
and y lie between 1 and ( p�1) and neither should have any factor in
common with ( p�1). In particular, since ( p�1) is even, neither x nor y
should be even. X and Y do not reveal their secret numbers to each
other or to anyone else.
(3) X computes the number
k x �m x (mod p)
and sends it to Y who raises it to the power y, giving the number (k x ) y .
Y computes the number
k y �m y (mod p)
and sends it to X who raises it to the power x, giving the number (k y ) x .