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 (/sup 1//spl sol//sub K/H(X/sup K//spl verbar/W), (/sup 1//spl sol//sub K/H(Y/sup K//spl verbar/W)) or /sup 1//spl sol//sub K/H(X/sup K/Y/sup K//spl verbar/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 C/sub 1/(X;Y) and C/sub 2/(X;Y), are considered. C/sub 1/(X;Y) is defined as the rate of the attainable minimum core of (X/sup K/,Y/sup K/) by removing each private information from (X/sup K/,Y/sup K/) as much as possible, while C/sub 2/(X;Y) is defined as the rate of the attainable maximum core V/sub C/ such that if one loses V/sub C/, then each uncertainty of X/sup K/ and Y/sup K/ becomes H(V/sub C/). It is proved that C/sub 1/(X;Y)=I(X;Y) and C/sub 2/(X;Y)=min /spl lcub/H(X), H(Y)/spl rcub/. C/sub 1/(X;Y) justifies the author's intuitive feeling that the mutual information represents a common information of X and Y.>