CCi.l/.l/r.\7t.l/7(>.V rilEUKV OF SECRECY SYSTEMS 689 



These liniils arc llie best possible and the equkvcalioiis in ques- 

 tion can be either for key or message. 

 The total redundancy Ds for A' letters of message is defined by 



Ds = log G - IKM) 



where G is the total number of messages of length .V and H{M) is the un- 

 certainty in choosing one of these. In a secrecy system where the total 

 number of possible cryptograms is equal to the number of possible messages 

 of length .V, II{E) < log G. Consequently 



IIe(K) = ^(A') + II{M) - H(E) 



> H(K) - [log G - H(M)]. 



Hence 



H{K) - He(K) < Dy . 



This shows that, in a closed system, for example, the decrease in equivoca- 

 tion of key after .V letters have been intercepted is not greater than the 

 redundancy of X letters of the language. In such systems, which comprise 

 the majority of ciphers, it is only the existence of redundancy in the original 

 messages that makes a solution possible. 



Now suppose we have a pure system. Let the different residue classes of 

 messages be Ci , d , C3 , ■ ■ ■ ,Cr , and the corresponding set of residue classes 

 of cryptograms he Ci , Co , ■ ■ ■, Cr. The probability of each E in C[ is the 

 same: 



P{E) = E a member of C, 



where >p, is the number of dififerent messages in C, . Thus we have 



H(E) = -L ., '^'^ log ^ 



- -2: Pia log ^^ 



Substituting in our equation for He(K) we obtain: 

 Theorem 11: For a pure cipher 



H,{K) = //(A) + H(M) + £ P(C,) log ^^ . 



This result can be used to compute He{K) in certain cases of interest. 



