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A month after William and Elizebeth were married in 1917, America declared war and found itself without an adequate cryptology department in WWI. The cryptology department of Riverbank soon began receiving cryptograms from the U.S. government, almost all of which Friedman solved easily, including a book cipher in use by a Hindu faction that was attempting to gain independence from Britain during the war. Friedman testified at the trial that led to their conviction, although the more interesting event at the trial was when a defendant shot a friend who was testifying and a marshal shot the defendant from a balcony.
Friedman taught military cryptology officers at Riverbank during the war, and needing a cryptography textbook of sorts, Friedman wrote an eight part series on cryptology known as the Riverbank Publications. Each piece of the series is remarkable in the history of cryptology. They blaze new trails and shine with clarity unlike any other work seen. Friedman tackled a wide variety of ciphers, and a wide variety of different variants of each general cipher, all with clarity and ingenuity.
At 28, Friedman wrote his most important scholarly work on cryptology, entitled, 'The Index of Coincidence and Its Applications in Cryptography'. This piece introduced the applications of statistical tools into the use of solving cryptograms. Friedman computed a term that represented the probability of a pair of letters from two series of texts being the same. Since the chance of choosing a letter randomly is 1/26, and the chance of drawing the same letter from another alphabet randomly is also 1/26, the chance of drawing two letters that are the same is (1/26 * 1/26). Of course, that is only for one letter, and Friedman computed the probability of getting any pair of letters. For that equation, we add the chance of drawing a single letter 26 times, or 26 * (1/26 * 1/26) which equals .0385. This means that if two series of one hundred letters each were placed on top of each other, there would be about 4 instances of the same letter. The example below, which you can generate again and again, should show you how accurate this is.
Friedman also computed the probability of getting a pairing from normal plaintext. To calculate this number, he utilized the frequency distribution charts for plaintext and used the percentiles to replace the 1/26. The chance of finding an 'e', was 13/100 * 13/100 and likewise for each letters' own frequency. This number turned out to be .0667, or about 7 pairs per two one hundred series. The example below shows a general example, though to test it yourself you can type in your own text to see for yourself.
The Index of Coincidence, as Friedman and now all of the cryptology world calls it, made cryptanalysis of polyalphabetic ciphers much easier by allowing cryptanalysists to more quickly find the correct place to line up two messages and proceed with a Kerckhoff solution. Friedman's pioneering work into statistics and probability brought cryptanalysis into the world of higher math, where it still stands today.