The d'Agapeyeff Cipher, revisited...

I know, I did blog about this only three days ago: but science moves ever onwards, OK?

A nice email arrived from Robert Matthews, the author of an excellent page on the d'Agapeyeff Cipher: he mentioned that he had received an email in February 2006 from John Willemse in Holland, who had suggested a novel kind of transposition cipher based around a spiral:-

I'm in no way a cipher expert, but I am a very curious person and I was wondering if the positioning of the 14x14 digram table could have anything to do with a spiral. The reason I suspect this, is that a spiraling positioning of numbers have the property that each upperleft corner of such a spiral (when starting with zero in the center) is a perfect square number. I'll try to illustrate my point:

16 15 14 13 12
17 .4 .3 .2 11 ..
18 .5 .0 .1 10 ..
19 .6 .7 .8 .9 26
20 21 22 23 24 25

Starting from zero, and counting up, anti-clockwise, you will encounter a perfect square of each even number in the topleft corner. 196 is also such a number.

The '04' digram almost in the center could be a break point. If you 'break' after the zero and shift the 4 to the right, creating a new set of digrams, you end up with a set of digrams before the zero and a set after the zero. The set after the zero should probably be reversed, either the whole set or the individual digrams, to create a similar set as the first one (the digrams starting with higher digits and ending with lower digits).

You might then be able to construct a spiral like positioning, with the zero in the center or the zero obmitted. The first set might then be 'twisted' around it clockwise, and the second set anti-clockwise, possibly interweaving each other.

These are just some wild ideas, and I'm in no way capable of constructing and verifying such a table myself, but maybe it's something to investigate?

Willemse's idea is certainly interesting: but let's look again at the (derived) 14x14 layout. To recap: one of the reasons for suspecting that transposition is involved is that there are two sets of horizontal tripled letters (75 75 75 and 63 63 63), while one of the reasons for suspecting that it's not a 'matrix transpose' diagonal flip is that there are two sets of vertical tripled letters (81 81 81 and 82 82 82). That is, unless the plaintext sadistically contains a phrase like "SEPIA AARDVARK" (a phrase which, I'm delighted to note, Google believes currently appears nowhere else on the Internet).

75 62 82 85 91 62 91 64 81 64 91 74 85 84
64 74 74 82 84 83 81 63 81 81 74 74 82 62
64 75 83 82 84 91 75 74 65 83 75 75 75 93
63 65 65 81 63 81 75 85 75 75 64 62 82 92
85 74 63 82 75 74 83 81 65 81 84 85 64 85
64 85 85 63 82 72 62 83 62 81 81 72 81 64
63 75 82 81 64 83 63 82 85 81 63 63 63 04
74 81 91 91 84 63 85 84 65 64 85 65 62 94
62 62 85 91 85 91 74 91 72 75 64 65 75 71
65 83 62 64 74 81 82 84 62 82 64 91 81 93
65 62 64 84 84 91 83 85 74 91 81 65 72 74
83 83 85 82 83 64 62 72 62 65 62 83 75 92
72 63 82 82 72 72 83 82 85 84 75 82 81 83
72 84 62 82 83 75 81 64 75 74 85 81 62 92

From this, it seems that, yes, you could construct a large number of spiral transpositions without tripled letter sequences. Yet I'm not completely convinced by the idea that the 04 token is a good indicator for the centre of a spiral: from the substitution cipher angle, I'd be quite happy to tag that as a likely 'X' or 'Y' in the plaintext instead.

However, I would point out that if you examine the various diagonal transpositions of the 14x14 (i.e. reading through the 14x14 one diagonal line at a time), there is (unless I'm somehow mistaken) apparently only a single tripled letter in two of them, and that only over a line-break:-

75 62 82 85 91 62 91 64 81 64 91 74 85 84
64 74 74 82 84 83 81 63 81 81 74 74 82 62
64 75 83 82 84 91 75 74 65 83 75 75 75 93
63 65 65 81 63 81 75 85 75 75 64 62 82 92
85 74 63 82 75 74 83 81 65 81 84 85 64 85
64 85 85 63 82 72 62 83 62 81 81 72 81 64
63 75 82 81 64 83 63 82 85 81 63 63 63 04
74 81 91 91 84 63 85 84 65 64 85 65 62 94
62 62 85 91 85 91 74 91 72 75 64 65 75 71
65 83 62 64 74 81 82 84 62 82 64 91 81 93
65 62 64 84 84 91 83 85 74 91 81 65 72 74
83 83 85 82 83 64 62 72 62 65 62 83 75 92
72 63 82 82 72 72 83 82 85 84 75 82 81 83
72 84 62 82 83 75 81 64 75 74 85 81 62 92

All in all, Willemse's idea of a spiral transposition does seem intriguing: but perhaps a little more psychologically ornate than d'Agapeyeff would have considered necessary as an exercise for the reader. If I were actively looking for a solution to this cipher (which I'm not), I would instead start with the four basic diagonal transpositions of the 14x14, and see if they led anywhere interesting... you never know! :-)