Coding theorems for Shannon´s cipher system with correlated source outputs, and common information

Source coding problems are treated for Shannon´s (1949) cipher system with correlated source outputs (X,Y). Several cases are considered based on whether both X and Y, only X, or only Y must be transmitted to the receiver, whether both X and Y, only X, or only Y must be kept secret, or whether the security level is measured by (¹/_KH(X^K|W), (¹/_KH(Y^K|W)) or ¹/_K H(X^KY^K|W) where W is a cryptogram. The admissible region of cryptogram rate and key rate for a given security level is derived for each case. Furthermore, two new kinds of common information of X and Y, say C₁(X;Y) and C₂(X;Y), are considered. C₁(X;Y) is defined as the rate of the attainable minimum core of (X^K,Y^K) by removing each private information from (X^K,Y^K) as much as possible, while C₂(X;Y) is defined as the rate of the attainable maximum core V_C such that if one loses V_C , then each uncertainty of X^K and Y^K becomes H(V_C). It is proved that C₁(X;Y)=I(X;Y) and C₂(X;Y)=min {H(X), H(Y)}. C₁(X;Y) justifies the author´s intuitive feeling that the mutual information represents a common information of X and Y