COMMLMCATIOy THEORY OF SECRECY SYSTEMS 691 



For /> = 3, 9 = I, and for p = \, q = ^, He(K, N) has been calculated and 

 is shown in Fig. 6. 



14. The Equivoc.vtiox Cilvkacteristic for a "Ra.vdom" Cipher 



In the preceding section we have calculated the equivocation charac- 

 teristic for a simple substitution applied to a two-letter language. This is 

 about the simplest type of cipher and the simplest language structure pos- 

 sible, yet alread}' the formulas are so involved as to be nearly useless. What 

 are we to do with cases of practical interest, say the involved transforma- 

 tions of a fractional transposition system applied to EngHsh with its ex- 

 tremely comple.x statistical structure? This complexity itself suggests a 

 method of approach. Sufficiently complicated problems can frequently be 

 solved statistically. To facilitate this we define the notion of a "random" 

 cipher. 



We make the following assumptions: 



1. The number of possible messages of length N is T = 2'^"'"', thus Ro = 

 log2 G, where G is the number of letters in the alphabet. The number of 

 possible cr}^ptograms of length N is also assumed to be T. 



2. The possible messages of length A' can be divided into two groups: 

 one group of high and fairly uniform a priori probability, the second 

 group of negligibly small total probability. The high probability group 

 will contain S = 2^^^ messages, where R = H{M)/X, that is, R is 

 the entropy of the message source per letter. 



3. The deciphering operation can be thought of as a series of lines, as 

 in Figs. 2 and 4, leading back from each E to various M's. We assume 

 k different equiprobable keys so there will be k lines leading back from 

 each E. For the random cipher we suppose that the lines from each 

 E go back to a random selection of the possible messages. Actually, 

 then, a random cipher is a whole ensemble of ciphers and the equivoca- 

 tion is the average equivocation for this ensemble. 



The equivocation of key is defined by 



IIe(K) = E P(E)Pe{K) log Pe(K). 



The probability that exactly m lines go back from a particular E to the high 

 probability group of messages is 



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If a cryptogram with m such lines is intercepted the equivocation is log m. 



TYlT 



The probability of such a cryptogram is — — , since it can be produced by 



