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136 CHAPTER 8. POLYALPHABETIC CIPHERS
the other, offset by various amounts. Let’s do this for the “KIOV” ciphertext,
using the offsets or shifts of 1 through 6. Whenever there is a coincidence, we
mark it with an asterisk.
1: 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
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
*
2: 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
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
3: 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
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
* * * * * *
4: 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
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
*
5: 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
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
*
6: 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
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
* * * *
Obviously this is a very short ciphertext, so the number of coincidences is low,
no matter the shift. But it is striking that the shifts of 3 (the keylength)
and 6 (twice the keylength) have many more coincidences than the other shift
amounts.
For a longer ciphertext doing this “shift examination” would be quite time
consuming. So let’s think more carefully about these coincidences. A coincidence
will occur when the same letter occurs twice in the ciphertext. How likely
are coincidences Or, to be more precise, how likely is it that two randomly
chosen letters from the ciphertext are the same
This brings us near the field of Probability. One of the first things anyone
learns in probability is that the likelihood of something particular happening is
the number of ways that thing can occur divided by the number of total things
that can occur:
Probability that A occurs =
Number of ways A can occur
Number of ways anything can occur
For example, my birthday is February 25th. How likely is it that I was born
on a weekend Since I didn’t tell you what year I was born, the best guess you
can make is 2/7-ths. There are two ways I could have been born on a weekend