Cryptology - Unofficial St. Mary's College of California Web Site

8.2. THE MEASURE OF ROUGHNESS 141
Putting everything together,
M.R. = (#A − x)2 + (#B − x) 2 + · · · + (#Z − x) 2
N 2
= #A2 + #B 2 + · · · + #Z 2
N 2 − 2 N 2 N 2
26
N 2 + 26
N 2
= #A2 + #B 2 + · · · + #Z 2
N 2 − 2
26 + 1
26
= #A2 + #B 2 + · · · + #Z 2
N 2 − 1
26 .
This looks a lot like the index of coincidence. Are they related We start
with the formula for Φ:
#A(#A − 1) + #B(#B − 1) + · · · + #Z(#Z − 1)
Φ =
N(N − 1)
= #A2 + #A + #B 2 + #B + · · · + #Z 2 + #Z
(multiplying out)
N(N − 1)
(
#A 2 + #B 2 · · · + #Z 2) + (#A + #B + · · · + #Z )
=
(regrouping)
N(N − 1)
(
#A 2 + #B 2 · · · + #Z 2) + N
=
(the sum of the letters is N)
N(N − 1)
(
#A 2 + #B 2 · · · + #Z 2) N
=
+
(separating the fractions)
N(N − 1) N(N − 1)
= N ((
#A 2
N − 1 × + #B 2 · · · + #Z 2)
N 2 + 1 )
N
(factoring out
N
N−1 )
For a long ciphertext,
close to 0. So
N
N−1 will be very close to 1, and, likewise 1 N
will be very
Φ ≈ #A2 + #B 2 · · · + #Z 2
N 2 . (8.2)
Hopefully this last fraction looks familiar: it is the final form for M.R. except
for a 1 26
. Taking care of that, we (finally!) conclude
Φ ≈ M.R. + 1 26 . (8.3)
The Index of Coincidence is basically a measure of roughness of the frequency
table! This is how Φ is connected to polyalphabetic ciphers.
William Friedman wrote any number of books and pamplets on cryptography.
Of particular interest to people like us, trying to break polyalphabetic ciphers,
is The Index of Coincidence and Its Applications in Cryptography, Riverbank
Publications No 22., 1920. This, according to David Kahn, “must be
regarded as the most important single publication in cryptology.”