Communication Theory of Secrecy Systems - Network Research Lab

(4) Each message in Ci can be enciphered into each cryptogram in C ′ i by
exactly k
ϕi
different keys. Similarly for decipherment.
The importance of the concept of a pure cipher (and the reason for the
name) lies in the fact that in a pure cipher all keys are essentially the same.
Whatever key is used for a particular message, the a posteriori probabilities
of all messages are identical. To see this, note that two different keys applied
to the same message lead to two cryptograms in the same residue class,
say C ′ i. The two cryptograms therefore could each be deciphered by k
ϕi keys
into each message in Ci and into no other possible messages. All keys being
equally likely the a posteriori probabilities of various messages are thus
PE(M) =
P (M)PM(E)
P (E)
= P (M)PM(E)
M P (M)PM(E)
= P (M)
P (Ci)
where M is in Ci, E is in C ′ i and the sum is over all messages in Ci. If E
and M are not in corresponding residue classes, PE(M) = 0. Similarly it can
be shown that the a posteriori probabilities of the different keys are the same
in value but these values are associated with different keys when a different
key is used. The same set of values of PE(K) have undergone a permutation
among the keys. Thus we have the result
Theorem 4. In a pure system the a posteriori probabilities of various messages
PE(M) are independent of the key that is chosen. The a posteriori probabilities
of the keys PE(K) are the same in value but undergo a permutation
with a different key choice.
Roughly we may say that any key choice leads to the same cryptanalytic
problem in a pure cipher. Since the different keys all result in cryptograms
in the same residue class this means that all cryptograms in the same residue
class are cryptanalytically equivalent–they lead to the same a posteriori probabilities
of messages and, apart from a permutation, the same probabilities of
keys.
As an example of this, simple substitution with all keys equally likely is a
pure cipher. The residue class corresponding to a given cryptogram E is the
E. In
set of all cryptograms that may be obtained from E by operations TiT −1
k
this case TiT −1
k is itself a substitution and hence any substitution on E gives
another member of the same residue class. Thus, if the cryptogram is
then
E = X C P P G C F Q,
E1 = R D H H G D S N
E2 = A B C C D B E F
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