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7.7. THE KASISKI TEST 121
Kasiski’s idea was to not look directly for the length of the keyword, but to
instead look for multiples of that length. Suppose a sequence of letters appears
more than once in the plaintext. And suppose the plaintext and keyword are
such that the same letters of the keyword encipher the same letters of these
repeated bits. Then the resulting ciphertext would be the same.
Now turn this around. If there is a repetition of some group of letters in
the ciphertext, perhaps this was caused by the same plaintext letters being
enciphered by the same keyword letters. If so, then the keyword would have
had to repeat just the right number of times to fit both the first and second
appearances of the plaintext letters. The distance these appearances are apart
would then be a multiple of the keyword length. If we can find several such
repetitions, so several such distances, then we ought to be able to focus in on
the length of the keyword.
To give a short example, consider the following cipher, with every fifth letter
numbered ([Kahn], page 208):
5 10 15 20 25 30
K I O V I E E I G K I O V N U R N V J N U V K H V M G Z I A
There are two repetitions here, one of KIOV and one of NU. The second KIOV
starts at letter 10, the first at letter 1, so they occur a distance 9 apart. The
second NU starts at letter 20, the first at letter 14, a distance of 6. Let’s put this
information in a chart:
Repetition Start Positions Distance Factors
KIOV 1, 10 10 − 1 = 9 3 × 3
NU 14, 20 20 − 14 = 6 2 × 3
What number(s) are both 2 × 3 and 3 × 3 a multiple of 3. So, Kasiski would
conclude, the keyword must be 3 letters long.
In fact, the plaintext is To be or not to be. That is the question.
and by putting the repeating keyword underneath we see that KIOV came from
the combination of tobe and RUNR, and NU from th and UN:
plaintext
key
ciphertext
t o b e o r n o t t o b e t h a t i s t h e q u e s t i o n
R U N R U N R U N R U N R U N R U N R U N R U N R U N R U N
K I O V I E E I G K I O V N U R N V J N U V K H V M G Z I A
Summarizing gives Kasiski’s Test.