674 BELL SYSTEM TECHNICAL JOURNAL 



be the same as the M space, i.e. that the system be endomorphic. The 

 fractional transposition is as homogeneous as the ordinary transposition 

 without being endomorphic. The proper definition is the following: A cipher 

 T is pure if for every T, , Tj , Tf, there is a Ts such that 



TiT'^Tk = T, 



and every key is equally likely. Otherwise the cipher is mixed. The systems of 



Fig. 2 are mixed. Fig. 4 is pure if all keys are equally likely. 



Theorem 1: In a pure cipher the operations TJ Tj which transform the message 



space into itself form a group whose order is m, the number of 



diferent keys. 

 For 



T-j'TkT-^Ti = / 



so that each element has an inverse. The associative law is true since these 

 are operations, and the group property follows from 



TtTjTtTi = TZ'nT~kTi = T^'Ti 



using our assumption that T~i Tj ^ T~s Tk for some s. 



The operation T~i Tj means, of course, enciphering the message with key 

 j and then deciphering with key i which brings us back to the message space. 

 If T is endomorphic, i.e. the Ti themselves transform the space ^m into itself 

 (as is the case with most ciphers, where both the message space and the 

 cryptogram space consist of sequences of letters), and the Ti are a group and 

 equally likely, then T is pure, since 



TiTfTk = TiTr = Ts . 



Theorem 2: The product of two pure ciphers which commute is pure. 



For if T and R commute TiRj = RiTm for every i,j with suitable /, m, and 



TiRj{TkRl) TmRn — TiRjR i TI TmRn 



= RuRv Ru'TrT\ Tt 



= RhTg. 



The commutation condition is not necessary, however, for the product to 

 be a pure cipher. 



A system with only one key, i.e., a single definite operation Tx , is pure 

 since the only choice of indices is 



TiT~iTx = Ti . 



Thus the expansion of a general cipher into a sum of such simple trans- 

 formations also exhibits it as a sum of pure ciphers. 



An examination of the example of a pure cipher shown in Fig. 4 discloses 



