The Entropy Ratio: Quantitative Confidentiality for Trapdoor Computing

A cipher map is a total function on bit strings implementing a trapdoor approximation of a latent function: the untrusted machine evaluates it blindly; only the trusted machine, holding the secret, can encode inputs and decode outputs. We develop a quantitative confidentiality theory for cipher map systems. The central measure is the entropy ratio e = H/H*, a normalized form of Shannon leakage from the Quantitative Information Flow literature. The representation-uniformity parameter delta controls the ratio via the Fannes-Audenaert continuity inequality, reducing confidentiality engineering to the concrete problem of minimizing delta. We analyze noise injection, multiple representations, and encoding granularity as three levers, prove a compositional leakage theorem, and validate experimentally on 20 Newsgroups encrypted Boolean search.