680 BELL SYSTEM TECHNICAL JOURNAL 



given the cryptaiialyst no information.^ Any action of his which depends on 

 the information contained in the cryptogram cannot be altered, for all of 

 his probabilities as to what the cryptogram contains remain unchanged. On 

 the other hand, if the condition is not satisfied there will exist situations in 

 which the enemy has certain a priori probabilities, and certain key and 

 message choices may occur for which the enemy's probabilities do change. 

 This in turn may afifect his actions and thus perfect secrecy has not been 

 obtained. Hence the definition given is necessarily required by our intuitive 

 ideas of what perfect secrecy should mean. 



A necessary and sufficient condition for perfect secrecy can be found 

 as follows: We have by Bayes' theorem 



PMI) = ''-^^^ 



in which: 



P{M) = a priori probability of message M. 



Pm{E) = conditional probability of cryptogram E if message M is 

 chosen, i.e. the sum of the probabilities of all keys which pro- 

 duce cryptogram E from message M. 

 P{E) = probability of obtaining cryptogram E from any cause.' 

 Pe{M) = a posteriori probability of message M if cryptogram E is 



intercepted. 

 For perfect secrecy Pe{M) must equal P{M) for all E and all M. Hence 

 either P{M) = 0, a solution that must be excluded since we demand the 

 equality independent of the values of P{M), or 



P^(E) = P{E) 



for every M and E. Conversely if Pm{E) = P(E) then 



Pe(M) = P{M) 



and we have perfect secrecy. Thus we have the result: 



Theorem 6: A necessary and sufUcient condHion for perfect secrecy is thai 



P„{E) = P{E) 



for all M and E. That is, Pm{E) must be independent of M. 

 Stated another way, the total probability of all keys that transform Mi 



' A purist niiglit. object tfiat the enemy lias obtained some information in that he Icnows 

 a message was sent. This may be answered by having among the messages a "biank" 

 corresponding to "no message." If no message is originated the blank is enciphered and 

 sent as a cryptogram. Then even this modicum of remaining information is efiminated. 



