690 



BELL SYSTEM TECHNICAL JOURNAL 



13. Equivocation for Simple Substitution on a Two Letter Language 



We will now calculate the equivocation in key or message when simple 

 substitution is applied to a two letter language, with probabilities p and q 

 for and 1, and successive letters chosen independently. We have 



He{M) = He{K) = -E F{E)PE{K)\ogPE{K) 

 The probability that E contains exactly 5 O's in a particular permutation is: 



2 4 6 S 10 12 14- 16 18 20 



NUMBER OF LETTERS, N 

 Fig. 6 — Equivocation for simple substitution on two-letter language. 



and the a posteriori probabilities of the identity and inverting substitutions 

 (the only two in the system) are respectively: 



P^(0) = 



P'q^ 



{p'q^-o + qop^-') 



PM = 



pff->qs 



(psqN-s _^ qopff-') 



There are ( " ) terms for each .y and hence 



HM<, N) 



?(*■>•.«- 



log 



p»qff-t 

 (^p'qN-s _|_ q'pN-i>f 



