Cryptography, Information Theory, and Error-Correction. Aiden A. Bruen

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      The method of coincidences

      We will now use the second principle of “coincidences,” to find the length of the keyword. The sequence of plain text letters in positions 1 to , to , , to , etc., should be approximately the same, especially if is large. It follows that a similar result holds true for the corresponding sequences of cipher text letters. Thus, if we slide the cipher text along itself and count the number of coincidences (i.e. agreements) at each displacement, then on average the maximum number of coincidences will occur after an integer multiple of the keyword length (i.e. the max occurs for some , ). This technique can be illustrated using the following example: We will first determine the period and use it to determine the nature of the keyword from a cipher text passage:

VVHZK	UHRGF	HGKDK	ITKDW	EFHEV	SGMPH
KIUWA	XGSQX	JQHRV	IUCCB	GACGF	SPGLH
GFHHD	MHZGF	BSPSW	SDSXR	DFHEM	OEPGI
QXKZW	LGHZI	PHLIV	VFIPH	XVA

      To find , write the cipher text on a long strip of paper, and copy it onto a second strip of paper beneath the first strip. For a displacement of 1, move the bottom strip to the left by one letter and count the number of character coincidences. Repeat this process for several displacements until the maximum displacement is obtained. The shift shown below corresponds to a displacement of 3:

V

V

H

Z

K

U

H

R

G

F

H

G

K

D

V

V

H

Z

K

U

H

R

G

F

H

G

K

D

K

I

T

By repeating this process for a number of displacements, we obtain Table 2.2:

Table 2.2 Number of character coincidences corresponding to displacement

Displacement	Number of coincidences
1	4
2	4
3	9
4	12
5	5
6	2
7	7
8	7

      From our results, the maximum number of occurrences appears for a displacement of 4. Since we know the maximum displacement occurs for a scalar multiple of the period, the period is likely either 2 or 4.

      Remark

      In applying the second principle, we are using a probabilistic argument. That is, in the above example, we cannot be certain that the period is either 2 or 4; however, we can say with high probability that it is likely to be either 2 or 4. If we were unable to decipher the text with a keyword length of 2 or 4, we would then try with the next highest number of coincidences, which occurs for displacement 3.

      Finding the keyword

Now that we know how to find , we examine how to find the keyword itself. Suppose, for example, that the key‐length is 7. Consider the plain text character in the positions (i.e. characters at a distance of 7 spaces). If a particular letter occurs in positions 1 and 8, the cipher text letters in positions 1 and 8 will be the same (because we use the same key letter to encipher both characters). How can we deduce which cipher text characters correspond to which plain text characters? In the English language, the most frequently СКАЧАТЬ