672 BELL SYSTEM TECHNICAL JOURNAL 



It may be noted that multiplication is not in general commutative, (we 

 do not always have RS = SR), although in special cases, such as substitu- 

 tion and transposition, it is. Since it represents an operation it is definition- 

 ally associative. That is RiST) = (RS)T = RST. Furthermore we have 

 the laws 



pip'T + q'R) + qS = pp'T + pq'R + qS 



(weighted associative law for addition) 



T{pR + qS) = pTR + qTS 

 (pR + qS)T = pRT + qST 



(right and left hand distributive laws) 

 and 



piT + p.2T + p,R = (Pi + p-2)T + p:ji 



It should be emphasized that these combining operations of addition 

 and multiplication apply to secrecy systems as a whole. The product of two 

 systems TR should not be confused with the product of the transformations 

 in the systems TiRj , which also appears often in this work. The former TR 

 is a secrecy system, i.e., a set of transformations with associated prob- 

 abilities; the latter is a particular transformation. Further the sum of two 

 systems pR -\- qT is a. system— the sum of two transformations is not de- 

 fined. The systems T and R may commute without the individual Ti and Rj 

 commuting, e.g., if i^ is a Beaufort system of a given period, all keys equally 

 likely, 



RiRj dp RjRi 



in general, but of course RR does not depend on its order; actually 



RR = V 



the Vigenere of the same period with random key. On the other hand, if 

 the individual Ti and Rj of two systems T and R commute, then the sys- 

 tems commute. 



A system whose M and E spaces can be identified, a \'ery common case 

 as when letter sequences are transformed into letter sequences, may be 

 termed endomorphic. An endomorphic system T may be raised to a power T" . 



A secrecy system T whose product with itself is equal to T, i.e., for wliich 



TT = T, 



will be called idempotent. For example, simple substitution, transposition 

 of period p, Vigenere of period p (all with each key equally likely) are 

 idempotent. 



