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8.1. COINCIDENCES 137
(being born on Saturday or being born on Sunday) and a total of seven days of
the week that I could have been born on.
How does this work for us To make the explanation clear, let’s write #A to
mean the number of A’s in the ciphertext. So #B represents the number of B’s,
#C is the number of C’s, and so on. How many ways are there to choose two
different A’s from the ciphertext There are #A ways to pick one A, and then
#A−1 ways to pick a different A. (Minus 1 because we’ve already picked one, so
there are one fewer to choose from.) So there are #A(#A−1) ways to pick two
A’s. Likewise #B(#B−1) ways to pick two B’s, and #C(#C−1) ways to pick two
C’s. Doing this for all the letters in the ciphertext gives us
#A(#A − 1) + #B(#B − 1) + · · · + #Z(#Z − 1)
ways of having a coincidence. To see how likely a coincidence is we must divide
by the number of ways of choosing any two letters. If the total number of letters
in our ciphertext is N, then this number is N(N − 1). 1
Putting the pieces together, we 2 have reinvented Friedman’s famed Index
of Coincidence. Designated Φ (“phi”), this is the likelihood that two letters,
picked randomly from a ciphertext, are the same. As we’ve just determined, the
formula for Φ is
Φ =
#A(#A − 1) + #B(#B − 1) + · · · + #Z(#Z − 1)
, (8.1)
N(N − 1)
where #A is the number of A’s in the ciphertext, #B is the number of B’s in the
ciphertext, etc., and N = #A + #B + · · · + #Z is the total number of letters in
the ciphertext.
Friedman was, justifiably, quite proud of his Index. In the introduction
to The Index of Coincidence and Its Applications in Cryptography Riverbank
Publications No 22., 1920 he wrote “when such a treatment is possible, it is
one of the most useful and trustworthy methods in cryptography.” 3 However,
from our development, it is not quite clear what Φ tells us, or how to use it.
Clearly Φ measures, somehow, the frequency of coincidences in a polyalphabetic
cipher. But what does Φ = 0.045 mean To find out, we need to think about
the frequency counts in a different way.
1 (Actually, from n objects there are n(n − 1)/2 ways of choosing two of them: we must
divide by 2 because it doesn’t matter which one is chosen first and which is chosen second. So
the denominator and each term in the numerator should have included a “/2”. Fortunately,
all the two’s cancel out.)
2 Our presentation of these ideas borrows liberally from that of Abraham Sinkov’s book
Elementary Cryptanalysis. Sinkov (1907–1998) was one of the first three people hired by
Friedman to work in the Army’s Signal Intelligence Service. He headed the Communications
Intelligence Organization during World War II, the group largely in charge of intercepting and
breaking Japanese messages. His book was published in 1966 and is quite influential.
3 Kahn [Kahn, pg 376] tells the story that General Cartier of the French indexCartier,
General Cryptographic section saw Riverbank No. 22 and “thought so highly of it that he
had it translated and published forthwith – false-dating in “1921” to make it appear as if the
French work had come first!”