Table 1.1. The Polybius checkerboard.
1 2 3 4 5
1 a b c d e
2 f g h ij k
3 l m n o p
4 q r s t u
5 v w x y z
do is look in the appropriate table entry to get the letter (remembering, of
course, that 24 can be either an i or a j). For example, 22, 42, 15, 15, 43,
25, 11, 42, 15, 13, 34, 32, 24, 33, 22 decodes to either “Greeks are coming” or
“Greeks are comjng”; it’s clear from context that the first phrase is what’s
A cipher is a method of concealment in which the primary unit is a let-
ter. Letters in a message are replaced by other letters, numbers, or symbols,
or they are moved around to hide the order of the letters. The word cipher
is derived from the Arabic sifr, meaning nothing, and it dates back to the
seventh century BCE. We also use the word code, often interchangeably
with cipher, though there are differences. A code, from the Latin codex, is
a method of concealment that uses words, numbers, or syllables to replace
original words or phases. Codes were not used until much later. As the
Arabic culture spread throughout much of the western world during this
time, mathematics flourished and so too did secret writing and decryption.
This is when frequency analysis was first used to break ciphers (messages).
Frequency analysis uses the frequency of letters in an alphabet as a way
of guessing what the cipher is. For example, e and t are the two most com-
monly used letters in English, whereas a and k are the two most commonly
used letters in Arabic. Thus, “native language” makes a difference. Chap-
ters 4 and 5 include many examples of how frequency analysis can decrypt
Abu Yusef Ya’qab ibn ’Ishaq as-Sabbah al-Kindi (Alkindus to contem-
porary Europeans) was a Muslim mathematician, who lived in what is now
modern day Iraq between 801 and 873 AD. He was a prolific philosopher
and mathematician and was known by his contemporaries as “the Second
Teacher”, the first one being Aristotle [55]. An early introduction to work
at the House of Wisdom, the intellectual hub of the Golden Age of Islam,
brought him into contact with thousands of historical documents that were
to be translated into Arabic, setting him on a path of scientific inquiry few
were exposed to in that time [46].
Al-Kindi was the first known mathematician to develop and utilize the
frequency attack, a way of decrypting messages based on the relative
rarity of letters in a given language. The total of his work in this field was
published in his work On Deciphering Cryptographic Messages in 750 AD,
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