COMMUNICATION THEORY OF SECRECY SYSTEMS 687 



//k(A', S) < //k(/v, A^) s > n, 



IIk{M,S) < nK{M,N) S > N (If lor first .1 letters of text) 



//k(m, .V) < n,(K, .V) 



The qualification regarding .1 letters in the second result of the theorem 

 is so that the equivocation will not be calculated with respect to the amount 

 of message that has been intercepted. If it is, the message equivocation may 

 (and usually does) increase for a time, due merely to the fact that more 

 letters stand for a larger possible range of messages. The results of the 

 theorem are what we might hope from a good secrecy index, since we would 

 hardly expect to be worse ofi on the average after intercepting additional 

 material than before. The fact that they can be proved gives further justi- 

 lication to our use of the equivocation measure. 



The results of this theorem are a consequence of certain properties of con- 

 ditional entropy proved in MTC. Thus, to show the first or second state- 

 ments of Theorem 7, we have for any chance events A and B 



H{B) > Ha{B). 



If we identify B with the key (knowing the first S letters of cryptogram) 

 and .1 with the remaining N — S letters we obtain the iirst result. Similarly 

 identifying B with the message gives the second result. The last result fol- 

 lows from 



He{M) < He{K, M) = He{K) + He.k{M) 



and the fact that He.k(M) = since K and E uniquely determine M. 

 Since the message and key are chosen independently we have: 



H(M, K) = H{M) + H{K). 



P'urthermore, 



H{M, K) = H(E, K) - //(£) + ^^^(A'), 



the first equality resulting from the fact that knowledge of M and A' or of 

 E and A' is equivalent to knowledge of all three. Combining these two we 

 obtain a formula for the equivocation of key: 



He{K) = H{M) + H{K) - H{E). 



In particular, if H(M) = H(E) then the equivocation of key, He(K), is 

 equal to the a priori uncertainty of key, II (K). This occurs in the perfect 

 systems described above. 



A formula for the equivocation of message can be found by similar means. 

 We have: 



H(M, E) = H{E) + He{M) = H{M) + Hm{E) 



He{M) = H{M) + H^t{E) - H{E). 



