The I.C. or Index of Coincidence is an important concept in
the  determination of monoalphabeticity.

When we look at a distribution of letters we are testing the
"observed value of PHI= PHI(o) for the distribution against
the expected value of PHI = PHI(r).   The formulas for theses
are:

     PHI (r) = 0.0385N (N-1)

and for English Military text,

     PHI(p)  = 0.0667N (N-1)

      where M = the number of elements in the distribution.

The constant 0.0385 is the decimal equivalent of 1/26, i.e.,
the reciprocal of the number of elements in the alphabet.

The constant 0.0667 is the sum of the squares of the
probabilities of occurrence of the individual letters in the
English text.

The usual mathematical notation for expressing PHI (o) = the
sum from i=A to Z  of [fi (fi-1)]  for all integral values.
or fA(fA-1) +fB(fB-1).....fZ(fZ-1) summed.

The I.C. expresses the relative monalphabeticity based on a
comparison of the text under examination with the theoretical
I. C. of plain text.

The I.C. of the message is PHI(o)/ PHI(r)


The theoretical  I.C. of English plain text is 1.73, which is
the decimal equivalent of 0.0667/0.0385, the ratio of the
plain constant to the random constant.

The I.C. of pure random text is 0.0385/0.0385 = 1.00

Another way to think of the constant 0.0667 is as the 'repeat
rate for a language"


The I.C. changes as we increase the number of alphabets:



                    m            I.C.

                    1            0.0667
                    2            0.052
                    5            0.044
                   10            0.041
                   large         0.038

the last item approaches random text of 0.0385.

Both [FR1] and [SINK] have good discussions.  The question is
a good one.  I will revisit it in detail next round.  See
also [FR3] .

By the way the most famous cryptographic paper ever was
written by Friedman.  He defined the Index of Coincidence in
his Riverbank Publication number 22.  Get a copy from the
Library, it is worth the effort.

Stinson may be talking about a purely random alphabet or
0.0385/0.0385.  Stinson is not technically correct.  His
assumption fails when we expand the I.C. to reduce multi -
alphabet English or other texts.  The Stinson formula also
fails when we look at Aperiodic Substitution cases at the end
of this course.  There we must use another test for
probability called the X (chi) test.  Gleasons book on
Probability for the Cryptanalyst helps us with the theory.


LANAKI