How does one prove a claim about a highly sensitive object such as a nuclear weapon without revealing information about the object? This paradox has challenged nuclear arms control for more than five decades. We present a mechanism in the form of an interactive proof system that can validate the structure and composition of an object, such as a nuclear warhead, to arbitrary precision without revealing either its structure or composition. We introduce a tomographic method that simultaneously resolves both the geometric and isotopic makeup of an object. We also introduce a method of protecting information using a provably secure cryptographic hash that does not rely on electronics or software. These techniques, when combined with a suitable protocol, constitute an interactive proof system that could reject hoax items and clear authentic warheads with excellent sensitivity in reasonably short measurement times.