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We can write:

[2.36]

In the same way, as one has: H(Y, X) = H(X, Y), therefore:

[2.37]

In addition, one has the following inequalities:

[2.38]

and similarly:

[2.39]

SPECIAL CASES.–

– Noiseless channel: in this case, on receipt of yj, there is certainty about the symbol actually transmitted, called xi (one-to-one correspondence), therefore:

[2.40]

Consequently:

[2.41]

and:

[2.42]

– Channel with maximum power noise: in this case, the variable at the input is independent of that of the output, i.e.:

[2.43]

We then have:

[2.44]

[2.45]

[2.46]

Note.– In information security, if xi is the plaintext, and yj is the corresponding ciphertext, then p(xi/yi) = p(xi) is the condition of the perfect secret of a cryptosystem.