670 BELL SYSTEM TEL II MCA L JOURXAL 



1. Size of Key. 



The kcv must he transmitted In- ii()ii-in1('rcrptil)lc means from transmit- 

 ting to receivinf^ points. Sometimes it must be memorized. It is therefore 

 desirable to ha\-c the ke\- as small as possible. 



3. Comphwily of liticiphcring and Deciphering Operations. 



Enciphering^ and deciphering; should, of course, be as simple as j)ossil)le. 

 If they are done manuall>-, complexity leads to loss of time, errors, etc. If 

 done mechanically, complexity leads to large expensive machines. 



4. I^ropagaiion of Errors. 



In certain types of ciphers an error of one letter in enci{)hering or trans- 

 mission leads to a large number of errors in the deciphered text. The errors 

 are spread out by the deciphering operation, causing the loss of much in- 

 formation and frequent need for repetition of the cryptogram. It is naturally 

 desirable to minimize this error expansion. 



5. Expansion of Message. 



In some t\-pes of secrec}- systems the size of the message is increased by 

 the enciphering process. This undesirable effect may be seen in systems 

 where one attempts to swamp out message statistics by the addition of 

 many nulls, or where multiple substitutes are used. It also occurs in many 

 "concealment" types of systems (which are not usually secrecy systems in 

 the sense of our definition). 



6. The Algebr.a. of Secrecy Systems 



If we have two secrecy systems T and R we can often combine them in 

 various ways to form a new secrec>' system .V. If T and R ha\-e the same 

 domain (message space) we may form a kind of "weighted sum," 



S = pT + qR 



where p -\- q "^ 1- This operation consists of first making a preliminary 

 choice with probabilities p and q determining which of 7' and R is used, 

 'lliis choice is part of the key of .V. .\fter this is determined T or 7? is used as 

 originally defined. The total ke\' of .V must specif}- which of T and R is used 

 and which key of T (or R) is used. 



If T consists of the transformations 7\ , • • ■ , T,,, with i)robabilities />] , 

 • ■ • , />,„ and 7^ consists of 7?i . • • • , Ri with probal)ilities (/i , • •• , (/a- then S = 

 pT + ^7? consists of the transformations T^ , T^ , • ■ •, T„, , 7?i , • • •, 7?;. 

 with probabilities ppi , ppi > • • -, ppm , qq\ , qqi ^ • - -, qqi^ respectiveh-. 



More generally we can form the sum of a number of systems. 



