So given such an encrypting sequence, is there any easy way to do cryptanalysis and go backwards and work out the key?

It turns out that there is. For as the picture below demonstrates, in an additive cellular automaton like the one considered here the underlying rule is such that it allows one not only to deduce the form of a particular row from the row above it, but also to deduce the form of a particular column from the column to its right. And what this means is that if one has some segment of the encrypting sequence, corresponding to part of a column, then one can immediately use this to deduce the forms of a sequence of other columns, and thus to find the form of a row in the cellular automaton—and hence the original key.

But what happens if the encrypting sequence does not include every single cell in a particular column? One cannot then immediately use the method described above. But it turns out that the additive nature of the underlying rule still makes comparatively straightforward cryptanalysis possible.

The picture on the next page shows how this works. Because of additivity it turns out that one can deduce whether or not some cell a certain number of steps down a given column is black just by seeing whether there are an odd or even number of black cells in certain specific positions in the row at the top. And one can then immediately

An example of the basis for cryptanalysis of an additive cellular automaton. The first set of pictures show the ordinary evolution of the rule 60 cellular automaton, in which each successive row is deduced from the one above. The second set of pictures show a kind of sideways evolution in which the rule is reinterpreted so as to allow a column of cells to be deduced from the column immediately to its right. Note that in both cases the colors of cells in the area on the lower left cannot be determined without knowing the colors of more initial cells than are shown.

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