CO.\[.\fl'N/CAriO.\ THEORY OF S/'ICRF.CV SYSTEMS 715 



grams be Ci . Group willi Mi all messages that can l)e (;btainecl from i/| 

 by T~i^TjM\ , and call this class C\ . The same C\ would be obtained if we 

 started witii any other M in C\ since 



TsT^TiMi = TiMi. 



Similarly the same G would be obtained. 



Choosing an M not in C'l (if any such exist) we construct C> and C'^ in 

 the same way. Continuing in this manner we obtain the residue classes 

 with properties (1) and (2). Let Mi and M2 be in Ci and suppose 



M2 = TiTYMi. 



If El is in Ci and can be obtained from Mi by 



El = TaMi = TpMi =■••== T„Mi, 



then 



El = TaT'^TiMi = TpT~2'TiM. = • • • 

 - TxMo = T^Mi ■ ■ ■ 



Thus each Af , in d transforms into £1 by the same number of keys. Simi- 

 larly each Ei in Ci is obtained from any M in Ci by the same number of 

 keys. It follows that this number of keys is a divisor of the total number 

 of keys and hence we have properties (3) and (4). 



