692 BELL SYSTEM TECHNICAL JOURNAL 



T 



m keys from high probability messages each with probability — . Hence the 



equivocation is: 



We wish to find a simple approximation to this when k is large. If the 

 expected value of m, namely m — Sk/T, is » 1, the variation of log m 

 over the range where the binomial distribution assumes large values will 

 be small, and we can replace log m by log m. This can now be factored out 

 of the summation, which then reduces to m. Hence, in this condition, 



He{K) = log ^ = log ^ - log r + log k 



He{K) = H{K) - DN, 



where D is the redundancy per letter of the original language {D = Ds/N). 

 If m is small compared to the large k, the binomial distribution can be 

 approximated by a Poisson distribution: 



/A 



.m k—m , '■' A 



p q = — - 

 ml 



where X = — . Hence 



4 00 * wi 



HeiK) - z- e~^ ^ —. m log m. 



A 2 ml 



If we replace mhy m -\- 1, we obtain: 



00 X"* 



He{K) = e-^ y — , log (m + 1). 

 ■^ ml 



This may be used in the region where X is near unity. For X <$C 1, the only 

 important term in the series is that for m = 1; omitting the others we have: 



Hb(K) = e-^\ log 2 

 = X log 2 

 = 2-A-^yfe log 2 . 



To summarize: He(K), considered as a function of A'', the number of 



intercepted letters, starts off at H(K) when AT" = 0. It decreases linearly 



H(K) 

 with a slope —D out to the neighborhood of TV = . After a short 



transition region, Hb{K) follows an exponential with "half life" distance 



