700 BELL SYSTEM TECHNICAL JOURNAL 



Theorem 12: A necessary and sufficient condition that T be strongly ideal is 

 that, for any two keys, TJ Tj is a measure preserving transforma- 

 tion of the message space into itself. 

 This is true since the a posteriori probabiHty of each key is equal to its 



a priori probability if and only if this condition is satisfied. 



18. Ex.\MPLEs OF Ideal Secrecy Systems 



Suppose our language consists of a sequence of letters all chosen inde- 

 pendenth' and with equal probabilities. Then the redundancy is zero, and 

 from a result of section 12, He(K) = H(K). We obtain the result 

 Theorem 13: If all letters are equally likely and independent any closed cipher 

 is strongly ideal. 



The equivocation of message will rise along the key appearance char- 

 acteristic which will usually approach H(K), although in some cases it 

 does not. In the cases of w-gram substitution, transposition, Vigenere, and 

 variations, fractional, etc., we have strongly ideal systems for this simple 

 language with He{M) -^ H(K) as iV -^ <x> . 



Ideal secrecy systems sufifer from a number of disadvantages. 



1. The system must be closely matched to the language. This requires 

 an extensive study of the structure of the language by the designer. Also a 

 change in statistical structure or a selection from the set of possible mes- 

 sages, as in the case of probable words (words expected in this particular 

 cryptogram), renders the system vulnerable to analysis. 



2. The structure of natural languages is extremely comphcated, and this 

 implies a complexity of the transformations required to eliminate redun- 

 dancy. Thus any machine to perform this operation must necessarily be 

 quite involved, at least in the direction of information storage, since a 

 "dictionary" of magnitude greater than that of an ordinary^ dictionary is 

 to be expected. 



3. In general, the transformations required introduce a bad propagation 

 of error characteristic. Error in transmission of a single letter produces a 

 region of changes near it of size comparable to the length of statistical effects 

 in the original language. 



19. Further Remarks on Equivocation and Redundancy 



We have taken the redundancy of "normal English" to be about .7 deci- 

 mal digits per letter or a redundancy of 50%. This is on the assumption 

 that word divisions were omitted. It is an approximate figure based on sta- 

 tistical structure extending over about 8 letters, and assumes the text to 

 be of an ordinary type, such as newspaper writing, literary work, etc. We 

 may note here a method of roughly estimating this number that is of some 

 cryptographic interest. 



