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138 CHAPTER 8. POLYALPHABETIC CIPHERS
8.2 The Measure of Roughness
It is not necessarily clear how the Index of Coincidence is connected to polyalphabetic
ciphers. To glimpse this connection we must think back to the differences
between the frequencies count of monoalphabetic and polyalphabetic
ciphers. We have learned to recognize a Caesar Cipher from the intense highs
and lows of its frequency count: there are many letters that occur quite often
(the ciphertext versions of the etaoinshr letters) and many that seldom occur
(the uvwxyz-types). However, in polyalphabetic ciphertexts the frequencies are
less sharp; the highs are lower and the lows are higher. We might say that a
Caesar Cipher has a frequency count that is much rougher than the frequency
count of a Vigenère Cipher. Further, the longer the keyword of a Vigenère
Cipher is, the smoother the frequency count is.
To illustrate with an example, I’ve enciphered the quote of General Givierge
from Section 7.6 using keys of various lengths. (I used the alphabet as the key, so
the five letter key was ABCDE.) The frequency counts of the resulting ciphertexts
appear in Figure 8.1. Notice that as the keys get longer, there are fewer numbers
keylength 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
one 13 1 7 10 20 4 3 4 14 0 1 9 5 19 21 3 0 11 4 16 5 1 3 1 8 0
three 3 6 6 7 15 10 8 4 6 7 4 5 3 10 13 17 9 4 6 9 11 6 2 2 5 5
five 5 4 5 8 10 5 12 7 7 7 3 5 10 7 12 10 7 12 8 4 8 9 7 6 3 2
ten 10 5 7 5 7 5 6 4 6 5 4 10 6 11 13 4 6 13 6 7 9 6 10 7 3 8
twenty 10 11 9 10 11 7 8 5 7 8 5 8 6 6 6 5 5 6 4 3 8 6 7 9 7 6
Figure 8.1: Frequency Counts: Same quote, different keylengths.
that are much larger or much smaller than “average” and more that are in the
middle. In general, the longer a keyword is, the smoother the frequencies are,
and the shorter the keyword, the rougher the frequencies, which makes sense.
After all, the point of polyalphabetic ciphers was to have each plaintext letter
become many different cipherletters, causing the individual frequency counts to
become more and more similar.
Does the converse hold Can we somehow measure the “roughness” of a
frequency count, and then use the measurement to estimate the keylength
Let’s start by thinking about measuring roughness.
Example: Consider the three sets of numbers
{3, 3, 3, 3} {4, 0, 4, 4} {1, 4, 1, 6}.
We can probably all agree that all these sets move from “smoothest” to “roughest.”
Why do we feel so The latter two sets are rougher because their numbers
are more spread out, are farther away from each other. What mathematical
device can measure this