The d'Agapayeff Cipher, continued...

In a recent post, I mentioned the idea that the d'Agapeyeff cipher might involve a diagonal transposition on the 14x14 grid cryptologists suspect it may well have been based upon. To test this out a bit, I wrote a short C++ program (which I've uploaded here) which turns the number pairs into characters (for convenience) and prints out all four diagonal transpositions (forward, reverse, forward boustrophedon, reverse boustrophedon) starting from each of the four corners.

Because the number of doubled and tripled letters is a simple measure of whether a transposition is likely to be plausible or not, I counted those up as well. The next metric to calculate would be the unique letter adjacency count (i.e. how many unique pairs of letters appear for each ordering)... but that's a task for another day.

Interestingly, transpositions starting from the top-left corner (and their reverse-order reflections in the bottom-right corner) have no triple-letters at all, as well as far fewer double-letters (9/10/11 compared to 13/14/15) than transpositions that start from the top-right. Though intriguing, I don't know if this is statistically significant: I haven't determined what the predicted doublet and triplet count would be for a totally randomised transposition, perhaps calculating that too that would be a good idea.

For any passing cryptologers, here is the ASCII version of the d'Agapeyeff cipher (as output by the C++ code) when arranged as a 14x14 grid (in numerical order but without J), followed by the 16 diagonal transpositions with their associated double & triple counts. My guess is that the top left corner reverse diagonal transposition (the second one down, starting "KBDMIDPIK...") is most likely to be the correct transposition, but we shall see (hopefully!) if this is true...

K B M P Q B Q D L D Q I P O
D I I M O N L C L L I I M B
D K N M O Q K I E N K K K S
C E E L C L K P K K D B M R
P I C M K I N L E L O P D P
D P P C M G B N B L L G L D
C K M L D N C M P L C C C Y
I L Q Q O C P O E D P E B T
B B P Q P Q I Q G K D E K F
E N B D I L M O B M D Q L S
E B D O O Q N P I Q L E G I
N N P M N D B G B E B N K R
G C M M G G N M P O K M L N
G O B M N K L D K I P L B R

*** Top left corner ***
Forward order...
KDBDIMCKIPPENMQDIEMOBCPCLONQIKPMCQLDBLMCKLKCLEBQLMIKILDE
NPQDGNPELQNBBQONBLKNIIGNDDPCCNEKKIPGCPOIQPMBLDKMOOMMOLIO
PLOBKBBMNQMQELLPMSMGDNOGDCGDRNGBPBKPCLPKNGIMDECDLMBQDEBY
DPELQKTKOBELFIKNGSPMKILLRBNR
--> number of doubles = 11, number of triples = 0
Reverse order...
KBDMIDPIKCQMNEPBOMEIDQNOLCPCDLQCMPKILCKLKCMLBDLIKIMLQBEQ
LEPNGDQPNEIINKLBNOQBBNPIKKENCCPDDNGOMKDLBMPQIOPCGBKBOLPO
ILOMMOSMPLLEQMQNMBRDGCDGONDGMPLCPKBPBGNDCEDMIGNKYBEDQBML
TKQLEPDFLEBOKSGNKIIKMPRLLNBR
--> number of doubles = 9, number of triples = 0
Simple boustrophedon (forward then reverse)...
KBDDIMPIKCPENMQBOMEIDCPCLONQDLQCMPKIBLMCKLKCLDLIKIMLQBEE
NPQDGNPELQIINKLBNOQBBNGNDDPCCNEKKIPOMKDLBMPQIOPCGOMMOLIO
PLOBKBSMPLLEQMQNMBMGDNOGDCGDRPLCPKBPBGNKNGIMDECDYBEDQBML
DPELQKTFLEBOKIKNGSIKMPLLRNBR
--> number of doubles = 10, number of triples = 0
Reverse boustrophedon (reverse then forward)...
KDBMIDCKIPQMNEPDIEMOBQNOLCPCIKPMCQLDLCKLKCMLBEBQLMIKILDQ
LEPNGDQPNENBBQONBLKNIIPIKKENCCPDDNGGCPOIQPMBLDKMOBKBOLPO
ILOMMOBMNQMQELLPMSRDGCDGONDGMNGBPBKPCLPDCEDMIGNKLMBQDEBY
TKQLEPDKOBELFSGNKIPMKIRLLBNR
--> number of doubles = 9, number of triples = 0

*** Top right corner ***
Forward order...
OPBIMSQIKRDIKMPLLKBDDDLNDPLYQCEKOGCTBLIKLLCBFQNKPELCEKSP
OQKLBLPELIMMOLNNPDDQGRBIMCIBMEKDEKNKINLKGCOGMLNLRDKEMMNP
QBQBMBDECCDCIOIEKLCIPLOQMPBOPPPMQPLNGPIDKQQIQBMKCLPDODND
IBBONGLBNDMGKEBPMNENMMNCBGOG
--> number of doubles = 14, number of triples = 2
Reverse order...
OBPSMIRKIQPMKIDDDBKLLYLPDNLDTCGOKECQFBCLLKILBSKECLEPKNQI
LEPLBLKQOPRGQDDPNNLOMMNKEDKEMBICMIBRLNLMGOCGKLNIKBMBQBQP
NMMEKDLKEIOICDCCEDPOBPMQOLPICIPGNLPQMPPKMBQIQQKDDNDODPLC
LGNOBBIKGMDNBNMPBEMMNEBCNOGG
--> number of doubles = 15, number of triples = 1
Simple boustrophedon (forward then reverse)...
OBPIMSRKIQDIKMPDDBKLLDLNDPLYTCGOKECQBLIKLLCBFSKECLEPKNQP
OQKLBLPELIRGQDDPNNLOMMBIMCIBMEKDEKNRLNLMGOCGKLNIKDKEMMNP
QBQBMBLKEIOICDCCEDCIPLOQMPBOPIPGNLPQMPPDKQQIQBMKDNDODPLC
IBBONGLKGMDNBEBPMNMMNENCBOGG
--> number of doubles = 13, number of triples = 0
Reverse boustrophedon (reverse then forward)...
OPBSMIQIKRPMKIDLLKBDDYLPDNLDQCEKOGCTFBCLLKILBQNKPELCEKSI
LEPLBLKQOPMMOLNNPDDQGRNKEDKEMBICMIBKINLKGCOGMLNLRBMBQBQP
NMMEKDDECCDCIOIEKLPOBPMQOLPICPPMQPLNGPIKMBQIQQKDCLPDODND
LGNOBBIBNDMGKNMPBEENMMBCNGOG
--> number of doubles = 14, number of triples = 0

*** Bottom right corner ***
Forward order...
RNBRLLIKMPSGNKIFLEBOKTKQLEPDYBEDQBMLDCEDMIGNKPLCPKBPBGNR
DGCDGONDGMSMPLLEQMQNMBBKBOLPOILOMMOOMKDLBMPQIOPCGPIKKENC
CPDDNGIINKLBNOQBBNQLEPNGDQPNEDLIKIMLQBELCKLKCMLBDLQCMPKI
QNOLCPCBOMEIDQMNEPPIKCMIDBDK
--> number of doubles = 11, number of triples = 0
Reverse order...
RBNLLRPMKIIKNGSKOBELFDPELQKTLMBQDEBYKNGIMDECDNGBPBKPCLPM
GDNOGDCGDRBMNQMQELLPMSOMMOLIOPLOBKBGCPOIQPMBLDKMOGNDDPCC
NEKKIPNBBQONBLKNIIENPQDGNPELQEBQLMIKILDBLMCKLKCLIKPMCQLD
CPCLONQDIEMOBPENMQCKIPDIMDBK
--> number of doubles = 9, number of triples = 0
Simple boustrophedon (forward then reverse)...
RBNRLLPMKISGNKIKOBELFTKQLEPDLMBQDEBYDCEDMIGNKNGBPBKPCLPR
DGCDGONDGMBMNQMQELLPMSBKBOLPOILOMMOGCPOIQPMBLDKMOPIKKENC
CPDDNGNBBQONBLKNIIQLEPNGDQPNEEBQLMIKILDLCKLKCMLBIKPMCQLD
QNOLCPCDIEMOBQMNEPCKIPMIDDBK
--> number of doubles = 10, number of triples = 0
Reverse boustrophedon (reverse then forward)...
RNBLLRIKMPIKNGSFLEBOKDPELQKTYBEDQBMLKNGIMDECDPLCPKBPBGNM
GDNOGDCGDRSMPLLEQMQNMBOMMOLIOPLOBKBOMKDLBMPQIOPCGGNDDPCC
NEKKIPIINKLBNOQBBNENPQDGNPELQDLIKIMLQBEBLMCKLKCLDLQCMPKI
CPCLONQBOMEIDPENMQPIKCDIMBDK
--> number of doubles = 9, number of triples = 0

*** Bottom left corner ***
Forward order...
GOGBCNMMNENMPBEKGMDNBLGNOBBIDNDODPLCKMBQIQQKDIPGNLPQMPPP
OBPMQOLPICLKEIOICDCCEDBMBQBQPNMMEKDRLNLMGOCGKLNIKNKEDKEM
BICMIBRGQDDPNNLOMMILEPLBLKQOPSKECLEPKNQFBCLLKILBTCGOKECQ
YLPDNLDDDBKLLPMKIDRKIQSMIBPO
--> number of doubles = 14, number of triples = 2
Reverse order...
GGONCBENMMEBPMNBNDMGKIBBONGLCLPDODNDDKQQIQBMKPPMQPLNGPIC
IPLOQMPBOPDECCDCIOIEKLDKEMMNPQBQBMBKINLKGCOGMLNLRBIMCIBM
EKDEKNMMOLNNPDDQGRPOQKLBLPELIQNKPELCEKSBLIKLLCBFQCEKOGCT
DLNDPLYLLKBDDDIKMPQIKRIMSPBO
--> number of doubles = 15, number of triples = 1
Simple boustrophedon (forward then reverse)...
GGOBCNENMMNMPBEBNDMGKLGNOBBICLPDODNDKMBQIQQKDPPMQPLNGPIP
OBPMQOLPICDECCDCIOIEKLBMBQBQPNMMEKDKINLKGCOGMLNLRNKEDKEM
BICMIBMMOLNNPDDQGRILEPLBLKQOPQNKPELCEKSFBCLLKILBQCEKOGCT
YLPDNLDLLKBDDPMKIDQIKRSMIPBO
--> number of doubles = 13, number of triples = 0
Reverse boustrophedon (reverse then forward)...
GOGNCBMMNEEBPMNKGMDNBIBBONGLDNDODPLCDKQQIQBMKIPGNLPQMPPC
IPLOQMPBOPLKEIOICDCCEDDKEMMNPQBQBMBRLNLMGOCGKLNIKBIMCIBM
EKDEKNRGQDDPNNLOMMPOQKLBLPELISKECLEPKNQBLIKLLCBFTCGOKECQ
DLNDPLYDDBKLLDIKMPRKIQIMSBPO
--> number of doubles = 14, number of triples = 0

The d'Agapeyeff Cipher...

Back in 1939, Alexander d'Agapeyeff wrote a tidy little book called "Codes and Ciphers" on cryptography history: though you can now buy it print-on-demand, cheap copies of the original book often come up on the various second-hand book aggregators (such as bookfinder.com), which is where I got my copy of the "Revised and reset" 1949 edition.

What is now generally understood is that d'Agapeyeff wasn't really a cryptographer per se: he had previously written a similar book on cartography for the same publisher, and so thought to tackle cryptography.

On the very last page of the text (p.144), d'Agapeyeff dropped in a little cipher challenge, saying "Here is a cryptogram upon which the reader is invited to test his skill."

75628 28591 62916 48164 91748 58464 74748 28483 81638 18174
74826 26475 83828 49175 74658 37575 75936 36565 81638 17585
75756 46282 92857 46382 75748 38165 81848 56485 64858 56382
72628 36281 81728 16463 75828 16483 63828 58163 63630 47481
91918 46385 84656 48565 62946 26285 91859 17491 72756 46575
71658 36264 74818 28462 82649 18193 65626 48484 91838 57491
81657 27483 83858 28364 62726 26562 83759 27263 82827 27283
82858 47582 81837 28462 82837 58164 75748 58162 92000

This modest little cryptogram, now known as "the d'Agapayeff Cipher", has somehow remained unbroken for 70 years, and is often to be found alongside the Voynich Manuscript on lists of cipher enigmas.

The first thing to note is that adjacent columns are formed alternately from 67890 and 12345 characters respectively: which is a huge hint that what we are looking at is (in part, at least) a grid cipher, where each pair of numbers gives a position in a grid. If so, then we can throw away the "patristrocat" spaces between the blocks of numbers and rearrange them as pairs.

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 00 0[0]

The first hint that the order of these might have been scrambled ('transposed') comes from the two sets of tripled letters: 75 75 75 and 63 63 63. Five centuries ago, even Cicco Simonetta and his Milanese cipher clerks knew that tripled letters are very rare (the only one in Latin is "uvula", 'little egg'). The second hint that this is a transposition cipher is the total number of characters (apart from the "00" filler at the end): 14x14. If we discard the filler & rearrange the grid we get:-

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

This is very probably the starting point for the real cryptography (though the presence of tripled characters in the columns implies that it probably isn't a simple "matrix-like" diagonal transposition. Essentially, it seems that we now have to solve a 14x14 transposition cipher and a 5x5 substitution cipher simultaneously, over a relatively small cryptogram - an immense number of combinations to explore.

However, we know that d'Agapeyeff wasn't a full-on cryptographer, so we should really explore the psychological angle before going crazy with an 800-year-long brute-force search. For a start, if you lay out the frequencies for the 5x5 letter grid (with 12345 on top, 67890 on the left), a pattern immediately appears:-

** .1 .2 .3 .4 .5
6. _0 17 12 16 11
7. _1 _9 _0 14 17
8. 20 17 15 11 17
9. 12 _3 _2 _1 _0
0. _0 _0 _0 _1 _0

Here, the 61 (top-left) frequency is 0, the 73 frequency is 0, and the final nine frequencies are 3, 2, 1, 0; 0, 0, 0, 1, 0. I think this points to a 5x5 mapping generated by a keyphrase, such as "Alexander d'Agapeyeff is cool" (for example). To make a keyphrase into a 5x5 alphabet, turn all Js into Is (say), remove all duplicate letters (and so it becomes ALEXNDRGPYFISCO), and then pad to the end with any unused characters in the alphabet in sequence (BHKMQTUVWZ)

* 1 2 3 4 5
6 A L E X N
7 D R G P Y
8 F I S C O
9 B H K M Q
0 T U V W Z

For a long-ish (but language-like) keyphrase, rare characters would tend to get moved to the end of the block: which is what we appear to see in the frequency counts above, suggesting that the final few letters are (for example) W X Y Z or W X Z.

Yet 61 and 73 have frequency counts of zero, which points to their being really rare letters (like Q or Z). However, if you read the frequency counts as strings, 61 62 63 = 0 17 12, while 73 74 75 = 0 14 17: which perhaps points to the first letter of the keyphrase (i.e. 61) being a rare consonant, and the second pair being Q U followed by a vowel. Might 73 74 75 76 77 be QUIET or QUITE?

I don't (of course) know: but I do strongly suspect that it might be possible for a cunning cryptographer to crack d'Agapeyeff's keyphrase quite independently of his transposition cipher. It can't be that hard, can it? ;-p

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Update: a follow-up post is here...