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8.3. THE FRIEDMAN TEST 143
The Friedman Test: Given a ciphertext, perhaps from a polyalphabetic
cipher, compute
Then
Φ =
#A(#A − 1) + #B(#B − 1) + · · · + #Z(#Z − 1)
.
N(N − 1)
1. If Φ is near .065, the cipher is likely to be a monoalphabetic one. If
Φ is closer to .038, the cipher is likely to be polyalphabetic.
2. The value of Φ to be expected from a polyalphabetic ciphers of given
keylength is
Keylength Expected Φ
1 .065601
2 .052036
3 .047515
4 .045254
5 .043898
6 .042994
10 .041186
large .038461
Suppose, for example, we compute that a cipher text has Φ = 0.47. Then
Friedman’s Test 7 would suggest that the keyword is probably three or four letters
long. Notice from the wiggle words (“suggest” and “probably”) that this method
gives only an approximation and there will frequently be cases in which the
Φ = .043 but the length of the keyword is actually 3. To double check that the
correct keylength was found one can either
1. Do a brief Kasiski Examination of the text to see if you can find 3 or 4
sequences of letters that support the Friedman computation. Or
2. Use the keylength suggest by the Friedman test to divide the ciphertext
into that number of rows. Recompute Φ for each row. These Φ’s will
be between .03 and .07. If most of them are closer to .07 than .03 you
have probably discovered the correct keylength. (It might seem time consuming
to recompute several mini−Φ’s, but since we will need to do a
frequency count for each of the rows anyway there is relatively little extra
computation.)
7 Calling this “The Friedman Test” is wildly unfair to Friedman. The Index of Coincidence
has many applications, only one of which is estimating the keylength of a polyalphabetic
cipher. Another application (see Exercises 8.20) is the determination of a ciphertext’s original
language. Another, more important, one is to solve the superimposition problem: given
several, perhaps partial, polyalphabetic ciphertexts with the same key, how should the texts
be lined up under one another (“superimposed”) so that the letters in each column have the
same keyletter. This is especially valuable when breaking machine ciphers of the type used in
the 1930’s and 1940’s.