Construction of a shared secret key using continuous variables

Motivated by recent advances in quantum cryptography with continuous variables, we study the problem of extracting a shared digital secret key from two correlated real values. Alice has access to a real value, X_A, and Bob to another value, X_B, such that I(X_A; X_B)>0. They wish to convert their values into a shared secret digital information while leaking as little information as possible to Eve. We show how the problem can be decomposed into two subproblems known in other contexts. The first is the design of a quantizer that maximizes a mutual information criterion, the second is known as coding with side information.