Classical Analog to Entanglement Reversibility

In this Letter we study the problem of secrecy reversibility. This asks when two honest parties can distill secret bits from some tripartite distribution p(XYZ) and transform secret bits back into p(XYZ) at equal rates using local operation and public communication. This is the classical analog to the well-studied problem of reversibly concentrating and diluting entanglement in a quantum state. We identify the structure of distributions possessing reversible secrecy when one of the honest parties holds a binary distribution, and it is possible that all reversible distributions have this form. These distributions are more general than what is obtained by simply constructing a classical analog to the family of quantum states known to have reversible entanglement. An indispensable tool used in our analysis is a conditional form of the Gacs-Korner common information.