encrypting sequence, and which are therefore encrypted using a particular element of the key. Then one can ask whether such squares are more often black or more often white, and one can compare this with the result obtained by looking at the frequencies of letters in the language of the original message. If these two results are the same, then it suggests that the corresponding element in the key is white, and if they are different then it suggests that it is black. And once one has found a candidate key it is easy to check whether the key is correct by trying to use it to recover some reasonably long part of the original message. For unless one has the correct key, the chance that what one recovers will be meaningful in the language of the original message is absolutely negligible.

So what happens if one uses a more complicated rule for generating an encrypting sequence from a key? Methods like the ones above still turn out to allow features of the encrypting sequence to be found. And so to make cryptography work it must be the case that even if one knows certain features or parts of the encrypting sequence it is still difficult to deduce the original key or otherwise to generate the rest of the sequence.

The picture below shows one way of generating encrypting sequences that was widely used in the early years of electronic cryptography, and is still sometimes used today. The basic idea is to look at the evolution of an additive cellular automaton in a register of limited width. The key then gives the initial condition for the cellular automaton, and the encrypting sequence is extracted, for example, by sampling a particular cell on successive steps.

Encryption using the rule 60 additive cellular automaton. This is essentially equivalent to a linear feedback shift register.

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From Stephen Wolfram: A New Kind of Science [citation]