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174 CHAPTER 9. DIGRAPHIC CIPHERS
9.3 Recognizing and Breaking Polygraphic Ciphers
What makes a digraphic cipher digraphic It encipher pairs of letters, rather
than single letters. For example, the word the will be broken up either as
*t-he or as th-e*. In a monoalphabetic cipher each plaintext letter is always
replaced by the same ciphertext letter. Don’t let the “di” in digraphic fool you –
digraphic ciphers also have this one-to-one replacement. Each pair of plaintext
letters is always replaced by the same pair of ciphertext letters. So there should
be many copies of the ciphertext versions of th and he in our ciphertext.
Further, and here is the key point, these repetitions will all occur at even
distances. If there are an odd number of letters between two occurrences of
the, then these two occurrences will be broken differently, and so will not lead
to a repetition. It is only when the is broken the same way that we will get
a repetition, and this only happens when there are an even number of letters
in between, that is, when the distance is even. So a cipher that is not monoalphabetic
but has many many digraph repetitions occurring at even distances is
almost certainly a digraphic cipher.
Let’s demonstrate with an example.
Example: Decrypt XCCNQ FUARE LXELM XSBUM WLKVO KTWJU EELXA NZUQJ
KCWFM KSKYN QOEYR QLXFK ELRKQ FYCSK OXELZ GZHQN UPIUM NYNVQ OXBRA VQAAN
IPRYJ YKOQO WXUMK JELOZ YSCII EJLXI MQAGJ FNIKO BJJMH EIURL RKQFR SLXSR
KJKOW FRQLX FKELR KEZ
Of course, this is a Hill cipher. But how might we determine this if we didn’t
know it As always, we start with a frequency count:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z total
6 3 5 0 12 8 2 2 7 9 16 13 7 8 9 2 12 11 6 1 8 3 5 11 7 5 178
This is not a Caesar cipher. Nor does it really look like a monoalphabetic
cipher, since the rare letters are not that rare. Is it a polyalphabetic cipher
Computing, Φ = 0.04621 suggests a keylength of about 4. In particular, this is
not a monoalphabetic cipher.
If it is polyalphabetic, we should be able to find some repetitions. Figure
9.4 contains all repetitions longer than length 2. The repetition RQLXFKELRK is
too long to be ignored: if this is a polyalphabetic cipher the keylength must
divide 112. But if we pick 7, 14, 28 or 56 as the keylength, we must ignore three
nice repetitions (LRKQF, RLFER, and QXKLK), all of whose distance is divisible by
22. This seems strange. Further, there are 57 repetitions of length 2 (which
we didn’t list). This is a huge number, far more than we’ve ever seen in a
polyalphabetic cipher. And almost all of them occur in lengths divisible by 2.
We have ruled out any of the ciphers we have ever seen, except for digraphic