COMMUNICATION THEORY OF SECRECY SYSTEMS 713 



F = LSLSLT 

 where T is a transposition, L is a linear operation, and 5 is a substitution. 



26. Ciphers of the Type TkFSj 



Suppose that F is a good mixing transformation that can be applied to 

 sequences of letters, and that Tk and Sj are any two simple families of trans- 

 formations, i.e., two simple ciphers, which may be the same. For concrete- 

 ness we may think of them as both simple substitutions. 



It appears that the cipher TFS will be a very good secrecy system from 

 the standpoint of its work characteristic. In the first place it is clear on 

 reviewing our arguments about statistical methods that no simple sta- 

 tistics will give information about the key — ^any significant statistics derived 

 from E must be of a highly involved and very sensitive type — the re- 

 dundancy has been both diffused and confused by the mixing transformation 

 F. Also probable words lead to a complex system of equations involving all 

 parts of the key (when the mix is good), which must be solved simultane- 

 ously. 



It is interesting to note that if the cipher T is omitted the remainmg 

 system is similar to S and thus no stronger. The enemy merely "unmixes" 

 the cryptogram by application of F-^ and then solves. If .S* is omitted the 

 remaining system is much stronger than T alone when the mix is good, but 

 still not comparable to TFS. 



The basic principle here of simple ciphers separated by a mixing trans- 

 formation can of course be extended. For example one could use 



TkFiSjFiRi 



with two mixes and three simple ciphers. One can also simplify by using the 

 same ciphers, and even the same keys as well as the same mixing transforma- 

 tions. This might well simplify the mechanization of such systems. 



The mixing transformation which separates the two (or more) appear- 

 ances of the key acts as a kind of barrier for the enemy — it is easy to carry 

 a known element over this barrier but an unknown (the key) does not go 

 easily. 



By supplying two sets of unknowns, the key for S and the key for T, 

 and separating them by the mixing transformation F we have "entangled" 

 the unknowns together in a way that makes solution very difficult. 



Although systems constructed on this principle would be extremely safe 

 they possess one grave disadvantage. If the mix is good then the propaga- 

 tion of errors is bad. A transmission error of one letter will affect several 

 letters on deciphering. 



