Perfect Secrecy and Kullback-Leibler divergence

50 Views Asked by At

Perfect Secrecy can be defined by

$I(M;Z^n) = D(P_{MZ^n}||P_MP_{Z^n}) = 0$

there the true joint distribution $P_{MZ^n}$ is compared to the product distributions $P_M P_{Z^n}$.

The system can be viewed as follows:

M -> Enc E -> X^n -> P_{yz|x} -> Y^n -> Dec -> ^M
                              -> Z^n -> Eve -> ?

I'm struggeling with the interpretation of the Kullback-Leibler divergence. Can someone give me some intuitive explanation?

1

There are 1 best solutions below

2
On

The KL-Divergence is defined as

$$ D(p_x || q_x) = \sum _x p(x) \log \frac{p(x)}{q(x)} $$

and the mutual information is defined as

$$ I(X;Y) = H(X) - H(X|Y) = \sum_x \sum_y p(x,y) \log \frac{p(x,y)}{p(x)p(y)} $$

from the above it's clear that $I(X;Y) = D(p_{xy} || p_xp_y )$