688 BELL SYSTEM TECHNICAL JOURNAL 



Tf wc have a product system S = TR, it is to be expected that the second 

 enciphering process will not decrease the equivocation of message. That this 

 is actually true can be shown as follows: Let M, Ex , Ei be the message and 

 the first and second encipherments, respectively. Then 



Pe,e,{M) = Pe,{M). 



Consequently 



He,e,{M) - He,{M). 



Since, for any chance variables, x, y, z, H^yiz) < Hy{z), we have the desired 

 result, He,{M) > He,{M). 



Theorem 8: The equivocation in message of a prodiid system S = TR is not 

 less than that when only R is used. 

 Suppose now we have a system T which can be written as a weighted sum 

 of several systems R, S, ■ ■ ■, U 



T ^ PiR+ PoS + ■■■ -\- p,nU E Z'. = 1 



and that systems R, S, ■ ■ ■ , U have equivocations Hi, H^, Hi, ■ ■ ■ , Hm. 



Theorem 9: The equivocation H of a weighted sum of systems is bounded 

 by the inequalities 



E prHi < II <1L p^ii^ - Z />- log p^ ■ 



These are best limits possible. The H's may be equivocations 

 either of key or message. 

 The upper limit is achieved, for example, in strongly ideal systems (to 

 be described later) where the decomposition is into the simple transforma- 

 tions of the system. The lower limit is achieved if all the systems R, S, 

 ■ ■ ■ , U go to completely different cryptogram spaces. This theorem is also 

 proved by the general inequalities governing equivocation, 



Ha{B) < H{B) < H{A) + H4B). 



We identify .1 with the particular system being used and B with the key 



or message. 



There is a similar theorem for weighted sums of languages. For this we 



identify A with the particular language. 



Theorem 10: Suppose a system can be applied to languages Li , L>, ■ ■ ■ , Lm 

 and has equivocation characteristics Hi, H-y, ■ ■ ■ , H,,, . When 

 applied to the iveighted sum JZ P^P^^ '/"' equivocation H is 

 bounded by 



Z P^H^ < li <11 P^H^ -HP^ log />. . 



