678 BELL SYSTEM TECHNICAL JOURNAL 



The columns are then cut apart and rearranged to make meaningful text. 

 When the columns are cut apart, the only information remaining is the 

 residue class of the cryptogram. 



Theorem 5: If T is pure then TiT'j T = T where TiTj are any two trans- 

 formations of T. Conversely if this is trice for any TiTj in a system 

 T then T is pure. 

 The first part of this theorem is obvious from the definition of a pure 

 system. To prove the second part we note first that, if TiTj T = T, then 

 TiTJ^Ts is a transformation of T. It remains to show that all keys are equi- 

 probable. We have T == 2J P^^s and 



s 



T.psT^T-'Ts - Y^PsTs. 



s s 



The term in the left hand sum with s = j yields pjTi . The only term in Ti 

 on the right is piTi . Since all coeSicients are nonnegative it follows that 



Pi < pi- 

 The same argument holds with i and/ interchanged and consequently 



pj = P^ 



and T is pure. Thus the condition that TiTJ^T = T might be used as an 

 alternative definition of a pure system. 



8. Similar Systems 



Two secrecy systems R and S will be said to be similar if there exists a 

 transformation A having an inverse A~^ such that 



R = AS 



This means that enciphering with R is the same as enciphering with S 

 and then operating on the result with the transformation A. If we write 

 RW S to mean R is similar to 5 then it is clear that R^=ii S implies S ^ R. 

 Also R ^ S and 5 ;^ T imply R ^ T and finally R ^ R. These are sum- 

 marized by saying that similarity is an equivalence relation. 



The cryptographic significance of similarity is that ii R ^ S then R and 

 5 are equivalent from the cryptanalytic point of view. Indeed if a cr}'pt- 

 analyst intercepts a cryptogram in system .S he can transform it to one in 

 system R by merely applying the transformation .1 to it. A cryptogram in 

 system R is transformed to one in .S' by applying .1^^ If R and S are ap- 

 plied to the same language or message space, there is a one-to-one correspond- 

 ence between the resulting cryptograms. Corresponding cryptograms give 

 the same distribution of a posteriori probabilities for all messages. 



If one has a method of breaking the system R then any system S similar 



