Primal Implication as Encryption

We propose a “cryptographic” interpretation for the propositional connectives of primal infon logic introduced by Y. Gurevich and I. Neeman and prove the corresponding soundness and completeness results. Primal implication φ→p ψ corresponds to the encryption of ψ with a secret key φ, primal disjunction φ ∨p ψ is a group key and ⊥ reflects some backdoor constructions such as full superuser permissions or a universal decryption key. For the logic of ⊥ as a universal key (it was never considered before) we prove that the derivability problem has linear time complexity. We also show that the universal key can be emulated using primal disjunction.