M IXING T RANSFORMATIONS

25 M IXING T RANSFORMATIONS

A notion that has proved valuable in certain branches of probability theory is the concept of a mixing transformation. Suppose we have a probability or measure space Ω and a measure preserving transformation F of the space into itself, that is, a transformation such that the measure of a transformed A notion that has proved valuable in certain branches of probability theory is the concept of a mixing transformation. Suppose we have a probability or measure space Ω and a measure preserving transformation F of the space into itself, that is, a transformation such that the measure of a transformed

F R 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 R n R approaches, as n→∞, the integral of the function over the entire space Ω 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 n R becomes a region consisting of a large number of thin filaments spread throughout Ω. 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 N as the space Ω 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 13 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 the folded again, etc.

In a good mixing transformation of a space with natural coordinates

i , with

X ′ 1 ,X 2 ,···,X S the point X i is carried by the transformation into a point X

i =f 1 (X 1 ,X 2 ,···,X S ) i = 1, 2, · · · , S and the functions f i are complicated, involving all the variables in a “sensi-

i consider- ably. If ′ X i passes through its range of possible variation the point X

tive” way. A small variation of any one, ′ X 3 , say, changes all the X

i 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 substitutions and simple linear operations, adding adjacent letters mod 26 for example. Thus we might take

13 E. Hopf, “On Causality, Statistics and Probability,” Journal of Math. and Physics, v. 13, pp. 51-102, 1934.

F = LSLSLT

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