Communication Theory of Secrecy Systems - Network Research Lab

in which E, M and K are the cryptogram, message and key and
P (E, K) is the probability of key K and cryptogram E
PE(K) is the a posteriori probability of key K if cryptogram E
is intercepted.
P (E, M) and PE(M) are the similar probabilities for message instead
of key.
The summation in HE(K) is over all possible cryptograms of a certain length
(say N letters) and over all keys. For HE(M) the summation is over all messages
and cryptograms of length N. Thus HE(K) and HE(M) are both functions
of N, the number of intercepted letters. This will sometimes be indicated
explicitly by writing HE(K, N) and HE(M, N). Note that these are
“total” equivocations; i.e., we do not divide by N to obtain the equivocation
rate which was used in MTC.
The same general arguments used to justify the equivocation as a measure
of uncertainty in communication theory apply here as well. We note that
zero equivocation requires that one message (or key) have unit probability, all
other zero, corresponding to complete knowledge. Considered as a function
of N, the gradual decrease of equivocation corresponds to increasing knowledge
of the original key or message. The two equivocation curves plotted as
functions of N, will be called the equivocation characteristics of the secrecy
system in question.
The values of HE(K, N) and HE(M, N) for the Caesar type cryptogram
considered above have been calculated and are given in the last row of Table
1. HE(K, N) and HE(M, N) are equal in this case and are given in decimal
digits (i.e., the logarithmic base 10 is used in the calculation). It should
be noted that the equivocation here is for a particular cryptogram, the summation
being only over M (or K), not over E. In general the summation
would be over all possible intercepted cryptograms of length N and would
give the average uncertainty. The computational difficulties are prohibitive
for this general calculation.
12 PROPERTIES OF EQUIVOCATION
Equivocation may be shown to have a number of interesting properties, most
of which fit into our intuitive picture of how such a quantity should behave.
We will first show that the equivocation of key or of a fixed part of a message
decreases when more enciphered material is intercepted.
Theorem 7. The equivocation of key HE(K, N) is a non-increasing function
of N. The equivocation of the first A letters of the message is a non-increasing
function of the number N which have been intercepted. If N letters have been
intercepted, the equivocation of the first N letters of message is less than or
equal to that of the key. These may be written:
686