The Shadow Knows: Refinement of Ignorance in Sequential Programs

Separating sequential-program state into “visible” and “hidden” parts facilitates reasoning about knowledge, security and privacy: applications include zero-knowledge protocols, and security contexts with hidden “high-security” state and visible “low-security” state. A rigorous definition of how specifications relate to implementations, as part of that reasoning, must ensure that implementations reveal no more than their specifications: they must, in effect, preserve ignorance. We propose just such a definition –a relation of ignorance-preserving refinement– between specifications and implementations of sequential programs. Its purpose is to enable a development-by-refinement methodology for applications like those above. Since preserving ignorance is an extra obligation, the proposed refinement relation restricts (rather than extends) the usual. We suggest general principles for restriction, and we give specific examples of them. To argue that we do not restrict too much –for “no refinements allowed at all” is trivially ignorance-preserving– we derive The Dining Cryptographers protocol via a program algebra based on the restricted refinement relation. It is also a motivating case study, as it has never before (we believe) been treated refinement-algebraically. In passing, we discuss –and solve– the Refinement Paradox.

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