Cryptology: SINKOV's Test for the Period

Cryptology

SINKOV's Test for the Period

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Definition

We want to test a pretended period l whether it is the real period. We write the text in rows of width l and consider the columns.

If l is the correct period, each column is monoalphabetically encrypted and has its coincidence index near the coincidence index of the plaintext language.
Otherwise the columns are random garbage and have coincidence indices near the random value 1/n. Or rather near the value for a polyalphabetic ciphertext of period (the true) l.

Maybe the columns are quite short, thus their coincidence indices are diffuse and give no clear impression. However we can put all the indices together without bothering about the different monoalphabets, and get a much more precise value, based on all the letters of the text.

Definition. The SINKOV statistic φ_l(a) of order l of a text a is the mean value of the coincidence indices over all l columns if the text is written in rows of width l.

Here is a Perl program that calculates the SINKOV statistics.

Example

Let us again examine the ciphertext from Section 3. We get the values:

φ₁(a)	0.0442	φ₇(a)	0.0829	φ₁₃(a)	0.0444
φ₂(a)	0.0439	φ₈(a)	0.0443	φ₁₄(a)	0.0839
φ₃(a)	0.0440	φ₉(a)	0.0427	φ₁₅(a)	0.0432
φ₄(a)	0.0438	φ₁₀(a)	0.0421	φ₁₆(a)	0.0439
φ₅(a)	0.0430	φ₁₁(a)	0.0426	φ₁₇(a)	0.0444
φ₆(a)	0.0435	φ₁₂(a)	0.0432	φ₁₈(a)	0.0419

The period 7 is overwhelmingly evident. The values other than at the multiples of 7 are in almost perfect compliance with a (German) ciphertext of period around 7.

Our example ciphertext was quite long, and it is no surprise that the statistical methods perform very well. To get a more realistic picture let us examine a ciphertext of length 148.

Author: Klaus Pommerening, 2013-Dec-23; last change: 2014-Feb-16.