Communication Theory of Secrecy Systems - Network Research Lab

These limits are the best possible and the equivocations in question can be
either for key or message.
The total redundancy DN for N letters of message is defined by
DN = log G − H(M)
where G is the total number of messages of length N and H(M) is the uncertainty
in choosing one of these. In a secrecy system where the total number of
possible cryptograms is equal to the number of possible messages of length
N, H(E)≤ log G. Consequently,
Hence
HE(K) = H(K) + H(M) − H(E)
≥ H(K) − [ log G − H(M)].
H(K) − HE(K) ≤ DN.
This shows that, in a closed system, for example, the decrease in equivocation
of key after N letters have been intercepted is not greater than the redundancy
of N letters of the language. In such systems, which comprise the majority of
ciphers, it is only the existence of redundancy in the original messages that
makes a solution possible.
Now suppose we have a pure system. Let the different residue classes of
messages be C1, C2, C3, · · · , Cr, and the corresponding set of residue classes
of cryptograms be C ′ 1, C ′ 2, C ′ 3, · · · , C ′ r. The probability of each E in C ′ 1 is the
same:
P (E) =
P (Ci)
ϕi
E a member of Ci
where ϕi is the number of different messages in Ci. Thus we have
H(E) = − P (Ci) P (Ci)
ϕi log
i
ϕi ϕi
= − P (Ci)
P (Ci) log
Substituting in our equation for HE(K) we obtain:
Theorem 11. For a pure cipher
i
HE(K) = H(K) + H(M) + P (Ci)
P (Ci) log .
This result can be used to compute HE(K) in certain cases of interest.
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i
ϕi
ϕi