Chameleon all-but-one trapdoor function (ABOTDF) is an important and useful primitive which was introduced in [9]. With the help of it, a more efficient blackbox construction of public key encryption (PKE) scheme, which is secure against chosen-ciphertext attack (CCA), can be given. In this paper, we formally generalize the construction of chameleon ABO-TDFs. As a special case of our generalization, a concrete construction of ABO-TDFs, which was first introduced by Peikert and Waters [1], is presented. Although the existence of lossy trapdoor functions is equivalent to that of ABO-TDFs by using the conversion in [1], as Peikert et al. said, the conversion involves some degradation in lossiness (i.e. additional leakage). Therefore, in this sense, our result is different from those in [21] where Hemenway et al. proved that homomorphic encryption with some additional properties implies lossy trapdoor functions.
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