COMMUNICATION rilEORY OF SECRECY SYSTEMS 



675 



certain properties. The messages fall into certain subsets which we will call 

 residue classes, and the possible cryi)tograms are divided into corresponding 

 residue classes. There is at least one line from each message in a class to 

 each cryptogram in the corresponding class, and no line between classes 

 which do not correspond. The number of messages in a class is a divisor of 

 the total number of keys. The number of lines "in parallel" from a message 

 M to a cryptogram in the corresponding class is equal to the number of 

 keys divided by the number of messages in the class containing the message 

 (or cryptogram). It is shown in the appendix that these hold in general for 

 pure ciphers. Summarized formally, we have: 



CRYPTOGRAM 



RESIDUE 



CLASSES 



] '' 



PURE SYSTEM 

 Fig. 4 — Pure system. 



Theorem 3: In a pure system the messages can be divided into a set of "residue 

 classes" Ci , C2 , ■ ■ ■ , Cs and the cryptograms into a corresponding 

 set of residue classes Ci , C2 , • ■ ■ ,Ca with the following properties: 



(1) The message residue classes are mutually exclusive and col- 

 lectively contain all possible messages. Similarly for the 

 cryptogram residue classes. 



(2) Enciphering any message in d with any key produces a 

 cryptogram in d . Deciphering any cryptogram in C,- zvith 

 any key leads to a message in d . 



(3) The number of messages in d , say <fi , is equal to the number 

 of cryptograms in d and is a divisor of k the number of keys. 



