COMMi'NICATlOX THEORY OF SECRECY SYSTEMS 671 



We note that any system T can be written as a sum of fixed operations 



T - P,T^ + p{r. + • • • + p,nl\n 



T, beinji; a defniile enciphering operation of T corres[)onding to key choice 

 /, which has probability p,. 



A second way of combining two secrecy systems is by taking the "prod- 

 uct," shown schematically in Fig. 3. Suppose T and R are two systems and 

 the domain (kmguage space) of R can be identified with the range (crypto- 

 gram space) of T. Then we can apply first 7' to our language and then R 



I 1 I 1 I 1 



t, — T — * — R » » •— ■ R"' -•— T" 



I > ' 



' ' 



Fig. 3 — Product of two systems S = RT. 



to the result of this enciphering process. This gives a resultant operation 5" 

 which we write as a product 



S = RT 



The key for .S" consists of both keys of 7' and R which are assumed chosen 

 according to their original probabilities and independently. Thus, if the 

 m keys of T are chosen with probabilities 



pi p2 ■ • • pm 



and the n keys of R have probabilities 



/ / / 



pi p2 • ■ ■ Pn , 



then .S' has at most mn keys with probabilities pipj . In many cases some of 

 the product transformaions RiTj will be the same and can be grouped to- 

 gether, adding their probabilities. 



Product encipherment is often used; for example, one follows a substi- 

 tution by a transposition or a transposition by a Vigenere, or applies a code 

 to the text and enciphers the result by substitution, transposition, frac- 

 tionation, etc. 



