In STOC 2008, Peikert and Waters introduced a powerful primitive called lossy trapdoor functions (LTFs). In a nutshell, LTFs are functions that behave in one of two modes. In the normal mode, functions are injective and invertible with a trapdoor. In the lossy mode, functions statistically lose information about their inputs. Moreover, the two modes are computationally indistinguishable. In this work, we put forward a relaxation of LTFs, namely, regularly lossy functions (RLFs). Compared to LTFs, the functions in the normal mode are not required to be efficiently invertible or even unnecessary to be injective. Instead, they could also be lossy, but in a regular manner. We also put forward richer abstractions of RLFs, namely all-but-one regularly lossy functions (ABO-RLFs). We show that (ABO)-RLFs admit efficient constructions from both a variety of number-theoretic assumptions and hash proof system (HPS) for subset membership problems satisfying natural algebraic properties. Thanks to the relaxations on functionality, the constructions enjoy shorter key size and better computational efficiency than that of (ABO)-LTFs. We demonstrate the applications of (ABO)-RLFs in leakage-resilient cryptography. - As a special case of RLFs, lossy functions imply leakage-resilient injective one-way functions with optimal leakage rate 1 - o(1). - ABO-RLFs immediately imply leakage-resilient message authentication code (MAC) with optimal leakage rate 1 - o(1), though in a weak sense. - ABO-RLFs together with HPS give rise to leakage-resilient chosen-ciphertext (CCA) secure key encapsulation mechanisms (KEM) (this approach extends naturally to the identity-based setting). Combining the construction of ABO-RLFs from HPS, this gives the first leakage-resilient CCA-secure public-key encryption (PKE) with optimal leakage rate based solely on HPS, and thus goes beyond the barrier posed by Dodis et al. (Asiacrypt 2010).
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