666 BELL SYSTEM TECHNICAL JOURNAL 



2. Transposition {Fixed Period d). 



The message is divided into groups of length d and a permutation appHed 

 to the first group, the same permutation to the second group, etc. The per- 

 mutation is the key and can be represented by a permutation of the first d 

 integers. Thus, for d = 5, we might have 2 3 1 5 4 as the permutation. 

 This means that: 



tn\ ni-i Ws nii ws me mi m^ m^ Wio • • ■ becomes 

 m« niz nil Ws rui m^ m%m^ mi(\ nig ■ ■ ■ . 



Sequential application of two or more transpositions will be called compound 

 transposition. If the periods are (/i , ^2 , • • •, d,, it is clear that the result is 

 a transposition of period d, where d is the least common multiple of di , 

 d2 , • • • , ds . 



3. Vigenere, arid Variations. 



In the Vigenere cipher the key consists of a series of d letters. These are 

 written repeatedly below the message and the two added modulo 26 (con- 

 sidering the alphabet numbered from .1 = to Z = 25. Thus 



ei — tUi -\- ki (mod 26) 



where ki is of period d in the index /. For example, with the key G A Hy 

 we obtain 



message N W I ST HE-- 



repeated key GAHGAHGA--- 



cryptogram TOD S AN E -■■ 



The Vigenere of period 1 is called the Caesar cipher. It is a simple substi- 

 tution in which each letter of M is advanced a fixed amount in the alphabet. 

 This amount is the key, which may be any number from to 25. The so- 

 called Beaufort and Variant Beaufort are similar to the Vigenere, and en- 

 cipher by the equations 



ei = k^— nii (mod 26) 



and 



ei = nii — ki (mod 26) 



respectively. The Beaufort of period one is called the reversed Caesar 

 cipher. 



The application of two or more Vigeneres in sequence will be called the 

 compound Vigenere. It has the equation 



ei = nti -\- ki+ li-\- --• -\- Si (mod 26) 



