COMMUNICATION THEORY OF SECRECY SYSTEMS 



695 



curves shown in Fig. ^). The key apj)earance characteristic in this case was 

 estimated by countin<r the number of different letters appearing in typical 

 English passages of A' letters. In so far as experimental data on the simple 

 substitution could be found, they agree very well with the curves of Fig. 9, 

 considering the various idealizations and approximations which have been 

 made. For example, the unicity point, at about 27 letters, can be shown 

 experimentally to lie between the limits 20 and 30. With 30 letters there is 



Graphical calculation of equivocation. 



nearly always a unique solution to a cryptogram of this type and with 20 

 it is usually easy to find a number of solutions. 



With transposition of period d (random key), H(K) = log dl, or about 

 d log d/e (using a Stirling approximation for </!). If we take .6 decimal digits 

 per letter as the appropriate redundancy, remembering the preservation of 

 letter frequencies, w^e obtain about l.7d log d/e as the unicity distance. 

 This also checks fairly well e.xperimentally. Note that in this case He{M) 

 is defined only for integral multiples of d. 



With the Vigenere the unicity point will occur at about Id letters, and 

 this too is about right. The Vigenere characteristic with the same key size 



