Lossy trapdoor relation and its applications to lossy encryption and adaptive trapdoor relation

Abstract

Peikert and Waters proposed the notion of lossy trapdoor function in STOC 2008. In this paper, we propose a relaxation of lossy trapdoor function, called lossy trapdoor relation. Unlike the lossy trapdoor function, lossy trapdoor relation does not require completely recovering the input but a public computable injective map of it. Interestingly, the lossy trapdoor relation maintains the application of lossy trapdoor function on the lossy encryption. Moreover, motivated by the construction of adaptive trapdoor relation proposed by Wee (Crypto 2010), we introduce all-but-one verifiable lossy trapdoor relation which is in fact a relaxation of all-but-one lossy trapdoor function. – The lossy trapdoor relation can be constructed from discrete logarithm related assumptions and subgroup membership assumptions efficiently. We also give an efficient construction of all-but-one verifiable lossy trapdoor relation from DLDH assumption over pairing group. As a byproduct, we propose an all-but-one lossy trapdoor function directly based on DLDH assumption which partially solve the open problem of Freeman et al. (PKC 2010). – The lossy trapdoor relation has a direct application to the lossy encryption and we propose new lossy encryptions based on three subgroup membership assumptions. The all-but-one verifiable lossy trapdoor relation can be used to construct adaptive trapdoor relation, which derives chosen ciphertext secure encryption.