More constructions of lossy and correlation-secure trapdoor functions

We propose new and improved instantiations of lossy trapdoor functions (Peikert and Waters in STOC'08, pp. 187-196, 2008), and correlation-secure trapdoor functions (Rosen and Segev in TCC'09, LNCS, vol. 5444, pp. 419-436, 2009). Our constructions widen the set of number-theoretic assumptions upon which these primitives can be based, and are summarized as follows: Lossy trapdoor functions based on the quadratic residuosity assumption. Our construction relies on modular squaring, and whereas previous such constructions were based on seemingly stronger assumptions, we present the first construction that is based solely on the quadratic residuosity assumption. We also present a generalization to higher-order power residues. Lossy trapdoor functions based on the composite residuosity assumption. Our construction guarantees essentially any required amount of lossiness, where at the same time the functions are more efficient than the matrix-based approach of Peikert and Waters. Lossy trapdoor functions based on the d-Linear assumption. Our construction both simplifies the DDH-based construction of Peikert and Waters and admits a generalization to the whole family of d-Linear assumptions without any loss of efficiency. Correlation-secure trapdoor functions related to the hardness of syndrome decoding.