COMMUNICATION THEORY OF SECRECY SYSTEMS 673 



The set of all endomorphic secrecy systems defined in a fixed message 

 space constitutes an "algebraic variety," that is, a kind of algebra, using 

 the operations of addition and multiplication. In fact, the properties of 

 addition and multiplication which we have discussed may be summarized 

 as follows: 



The set of endomorphic ciphers ivith the same message space and the two com- 

 bining operations of weighted addition and multiplication form a linear associ- 

 ative algebra idth a unit element, apart from the fad that the coefficients in a 

 weighted addition must be non-negative and sum to unity. 



The combining operations give us ways of constructing many new types 

 of secrecy systems from certain ones, such as the examples given. We may 

 also use them to describe the situation facing a cryptanalyst when attempt- 

 ing to solve a cryptogram of unknown type. He is, in fact, solving a secrecy 

 system of the type 



T = p,A + p.B^ •■■ + prS + p'X Z /» = 1 



where the ^, -B, ■ ■ ■ , S are known types of ciphers, with the pi their a priori 

 probabilities in this situation, and p'X corresponds to the possibility of a 

 completely new^ unknown type of cipher. 



7. Pure and Mixed Ciphers 



Certain types of ciphers, such as the simple substitution, the transposi- 

 tion of a given period, the Vigenere of a given period, the mixed alphabet 

 Vigenere, etc. (all with each key equally likely) have a certain homogeneity 

 with respect to key. Whatever the key, the enciphering, deciphering and 

 decrypting processes are essentially the same. This may be contrasted wath 

 the cipher 



pS+ qT 



where S is a simple substitution and T a transposition of a given period. 

 In this case the entire system changes for enciphering, deciphering and de- 

 cryptment, depending on whether the substitution or transposition is used. 

 The cause of the homogeneity in these systems stems from the group 

 {property— we notice that, in the above examples of homogeneous ciphers, 

 the product TiTj of any two transformations in the set is equal to a third 

 transformation Tf,- in the set. On the other hand TiSj does not equal any 

 transformation in the cipher 



pS + qT 



which contains only substitutions and transpositions, no products. 



We might define a "pure" cipher, then, as one whose Ti form a group. 

 This, however, would be too restrictive since it requires that the E space 



