The techniques of modern ‘digital’ communication have made familiar the idea that any piece of information, whether a written message, a photograph or even a sound can be transmitted in the form of a number. From this point of view a secret code consists of a finite subset U of the positive integers ℤ+ (the possible messages), a finite subset V of ℤ+ (the possible coded messages) together with a function T:U → V (the encoding function) and a function S:V→U (the decoding function) such that ST:U → U is the identity.

Remark. In fact even this definition fails to cover all possibilities since, for example, we could suppose T ‘multivalued’ with the value T(u) being chosen at random from a set Q(u)⊆Q such that ν∊Q(u) implies S(ν) = u. But we must start somewhere.

As a simple example let us take U = V = {n:0 ≤ n ≤ N − 1} and define T:U→V and S:V→U by the relations T(u) = u + M mod N, S(ν) = ν−M mod N. We consider the problem faced by ‘opponents’ who wish to decipher messages written using this code.

In general we must assume that our opponents know or guess the method of coding that we use. For the sake of illustration we may suppose that their information includes the value of N but not, at least initially, the value of M. Thus if we choose M at random and only use the code once, it is unbreakable, since trial decodes S′r(ν) = ν − r mod N allowing r to run from 0 to N − 1 will give all possible messages without any indication of which to choose.