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The existence of encryption schemes secure under Selective Opening Attack (SOA) has remained open despite considerable interest and attention. The authors provide the first public key encryption schemes secure against sender corruptions in this setting. The underlying tool is lossy encryption. The schemes have short keys. (Public and secret keys of a fixed length suffice for encrypting an arbitrary number of messages.) The schemes are stateless and non-interactive, and security does not rely on erasures. The schemes are without random oracles, proven secure under standard assumptions (DDH, Paillier's DCR, QR, lattices), and even efficient. They are able to meet both an indistinguishability (IND-SO-ENC) and a simulation-style, semantic security (SEM-SO-ENC) definition.
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