Building Lossy Trapdoor Functions from Lossy Encryption

Injective one-way trapdoor functions are one of the most fundamental cryptographic primitives. In this work we show how to derandomize lossy encryption (with long messages) to obtain lossy trapdoor functions, and hence injective one-way trapdoor functions. Bellare, Halevi, Sahai and Vadhan (CRYPTO ’98) showed that if Enc is an IND-CPA secure cryptosystem, and H is a random oracle, then x 7→ Enc(x,H(x)) is an injective trapdoor function. In this work, we show that if Enc is a lossy encryption with messages at least 1-bit longer than randomness, and h is a pairwise independent hash function, then x 7→ Enc(x, h(x)) is a lossy trapdoor function, and hence also an injective trapdoor function. The works of Peikert, Vaikuntanathan and Waters and Hemenway, Libert, Ostrovsky and Vergnaud showed that statistically-hiding 2-round Oblivious Transfer (OT) is equivalent to Lossy Encryption. In their construction, if the sender randomness is shorter than the message in the OT, it will also be shorter than the message in the lossy encryption. This gives an alternate interpretation of our main result. In this language, we show that any 2-message statistically sender-private semi-honest oblivious transfer (OT) for strings longer than the sender randomness implies the existence of injective one-way trapdoor functions. This is in contrast to the black box separation of injective trapdoor functions from many common cryptographic protocols, e.g. IND-CCA encryption.