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

142 CHAPTER 8. POLYALPHABETIC CIPHERS
“Before Friedman, cryptology eked out an existence as a study unto itself,
as an isolated phenomenon, neither borrowing from nor contributing to
other bodies of knowledge. ... It dwelt a recluse in the world of science.
Friedman let cryptology out of this lonely wilderness.” David Kahn [Kahn,
pg 383]
8.3 The Friedman Test
We have finished the preliminary work and can now start to exploit our formulas
to understand Φ. First, how large and how small can the values of Φ be
From Formula 8.3, Φ is large when M.R. is large. And M.R. will be large
when the frequency count is rougher. And the frequency count is roughest
for monoalphabetic ciphers. Our standard monoalphabetic frequencies counts
come from Figure 1.3. Using these numbers and Formula 8.2 (with N = 100
since the numbers are percentages) gives Φ = 0.065601 for monoalphabetically
enciphered ciphertexts. (See exercise 8.18.)
On the other hand, M.R. is always positive, and its smallest value is 0, when
all the numbers are the same. So from Formula 8.3, the smallest value of Φ is
0 + 1 26 ≈ .03846.
Do these values agree with our experimental data Adding Φ to Figure 8.1,
we have
length 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 Φ
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 0.0655738
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 0.0442563
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 0.0392122
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 0.0387318
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 0.0364499
A keylength of 1 has a Φ value of near .065, and as the keys get longer, the
value of Φ rapidly decreases to about 0.038.
Friedman did not stop here, however, but continued until he found a direct
relationship between the keylength and Φ. We won’t do that here (see exercise
8.19), but Friedman was able to show that
Φ ≈
.065(N − k) + .038N(k − 1)
,
k(N − 1)
where k is the keylength and the values .065 and .038 are those we found above.
Friedman then solved directly for k, giving us Friedman’s test.