712 BELL SYSTEM TECHNICAL JOURNAL 



transformed region FR is equal to the measure of the initial region R. The 

 transformation is called mixing if for any function defined over the space and 

 any region R the integral of the function over the region F"R approaches, 

 as w -^ 00 , the integral of the function over the entire space J2 multiplied 

 by the volume of R. This means that any initial region R is mixed with 

 uniform density throughout the entire space if F is applied a large number of 

 times. In general, F"R becomes a region consisting of a large number of thin 

 filaments spread throughout fi. As n increases the filaments become finer 

 and their density more constant. 



A mixing transformation in this precise sense can occur only in a space 

 with an infinite number of points, for in a finite point space the transforma- 

 tion must be periodic. Speaking loosely, however, we can think of a mixing 

 transformation as one which distributes any reasonably cohesive region in 

 the space fairly uniformly over the entire space. If the first region could be 

 described in simple terms, the second would require very complex ones. 



In cryptography we can think of all the possible messages of length A'^ 

 as the space fi and the high probability messages as the region R. This latter 

 group has a certain fairly simple statistical structure. If a mixing transforma- 

 tion were applied, the high probability messages would be scattered evenly 

 throughout the space. 



Good mixing transformations are often formed by repeated products of 

 two simple non-commuting operations. Hopf- has shown, for example, that 

 pastry dough can be mixed by such a sequence of operations. The dough is 

 first rolled out into a thin slab, then folded over, then rolled, and then 

 folded again, etc. 



In a good mixing transformation of a space with natural coordinates Xi , 

 X2 , • ■ • , Xs the point Xi is carried by the transformation into a point Xi , 

 with 



Xi = fi(X} , X2 , ■ ■ ■ , Xs) i = 1,2, ■ ■ • , S 



and the functions/,- are complicated, involving all the variables in a ''sensi- 

 tive" way. A small variation of any one, X3 , say, changes all the Xi con- 

 siderably. If X3 passes through its range of possible variation the point 

 Xi traces a long winding path around the space. 



Various methods of mixing applicable to statistical sequences of the type 

 found in natural languages can be devised. One which looks fairly good is 

 to follow a preliminary transposition by a sequence of alternating substi- 

 tutions and simple linear operations, adding adjacent letters mod 26 for 

 example. Thus we might take 



^ E. Hopf, "On Causality, Statistics and Probability," Journal of Math, and Physics, 

 V. 13, pp. 51-102, 1934. 



