698 BELL SYSTEM TECHNICAL JOURNAL 



as simple substitution will be approximately as shown in Fig. 10. The 

 Vigenere, Playfair and Fractional cases are more likely to follow the the- 

 oretical formulas for random ciphers than simple substitution and trans- 

 position. The reason for this is that they are more complex and give better 

 mixing characteristics to the messages on which they operate. 



The mixed alphabet Vigenere (each of d alphabets mixed independently 

 and used sequentially) has a key size, 



H(K) = J log 26! = 26.3^ 



and its unicity point should be at about 53d letters. 



These conclusions can also be put to a rough experimental test with the 

 Caesar type cipher. In the particular cryptogram analyzed in Table I, 

 section 11, the function (He(K, A^) has been calculated and is given below, 

 together with the values for a random cipher. 



N 

 H (observed) 

 H (calculated) 



The agreement is seen to be quite good, especially when we remember 

 that the observed H should actually be the average of many different cryp- 

 tograms, and that D for the larger values of iV is only roughly estimated. 



It appears then that the random cipher analysis can be used to estimate 

 equivocation characteristics and the unicity distance for the ordinary 

 types of ciphers. 



16. Validity of a Cryptogram Solution 



The equivocation formulas are relevant to questions which sometimes 

 arise in cryptographic work regarding the validity of an alleged solution 

 to a cryptogram. In the history of cryptography there have been many 

 cryptograms, or possible cryptograms, where clever analysts have found 

 a "solution." It involved, however, such a complex process, or the material 

 was so meager that the question arose as to whether the cryptanalyst had 

 "read a solution" into the cryptogram. See, for example, the Bacon-Shake- 

 speare ciphers and the "Roger Bacon" manuscript.^" 



In general we may say that if a proposed system and key solves a crypto- 

 gram for a length of material considerably greater than the unicity distance 

 the solution is trustworthy. If the material is of the same order or shorter 

 than the unicity distance the solution is highly suspicious. 



This effect of redundancy in gradually producing a unique solution to 

 a cipher can be thought of in another way which is helpful. The redundancy 

 is essentially a series of conditions on the letters of the message, which 



•0 See Fletcher Pratt, loc. cU. 



