Cipher Maps: Total Functions as Trapdoor Approximations

Privacy in encrypted computation need not come from access-pattern hiding (ORAM), exact algebraic homomorphism (FHE), or simulation-based security (garbled circuits). We show that a cipher map, a total function on bit strings where the concepts of domain, correctness, and function identity exist only behind a one-way trapdoor, provides a distinct, quantifiable privacy model. The entire construction reduces to a single design choice: an acceptance predicate that partitions hash space among output values. We formalize the framework through four measurable properties, prove the composition and space-optimality theorems, and demonstrate the construction on arbitrary maps, set membership, and encrypted search.