Editing Vigenère cipher (section)

===Friedman test===
The Friedman test (sometimes known as the kappa test) was invented during the 1920s by [[William F. Friedman]], who used the [[index of coincidence]], which measures the unevenness of the cipher letter frequencies to break the cipher. By knowing the probability <math>\kappa_\text{p}</math> that any two randomly chosen source language letters are the same (around 0.067 for [[Case sensitivity|case-insensitive]] English) and the probability of a coincidence for a uniform random selection from the alphabet <math>\kappa_\text{r}</math> ({{frac|1|26}} = 0.0385 for English), the key length can be estimated as the following:

: <math>\frac{\kappa_\text{p}-\kappa_\text{r}}{\kappa_\text{o}-\kappa_\text{r}}</math>

from the observed coincidence rate

: <math>\kappa_\text{o}=\frac{\sum_{i=1}^{c}n_i(n_i -1)}{N(N-1)}</math>

in which ''c'' is the size of the alphabet (26 for English), ''N'' is the length of the text and ''n''<sub>1</sub> to ''n''<sub>''c''</sub> are the observed ciphertext [[letter frequencies]], as integers.

That is, however, only an approximation; its accuracy increases with the length of the text. It would, in practice, be necessary to try various key lengths that are close to the estimate.<ref>{{cite book |editor=Henk C.A. van Tilborg |title=Encyclopedia of Cryptography and Security |url=https://archive.org/details/encyclopediacryp00tilb |url-access=limited |publisher=Springer |edition=First |isbn=0-387-23473-X |pages=[https://archive.org/details/encyclopediacryp00tilb/page/n127 115] |year=2005}}</ref> A better approach for repeating-key ciphers is to copy the ciphertext into rows of a matrix with as many columns as an assumed key length and then to compute the average [[Index of coincidence#Example|index of coincidence]] with each column considered separately. When that is done for each possible key length, the highest average index of coincidence then corresponds to the most-likely key length.<ref>{{cite journal | author=Mountjoy, Marjorie | title= The Bar Statistics | journal=NSA Technical Journal | year=1963 | volume=VII | issue=2,4}}  Published in two parts.</ref> Such tests may be supplemented by information from the Kasiski examination.