Compact lossy and all-but-one trapdoor functions from lattice

Lossy trapdoor functions (LTDF) and all-but-one trapdoor functions (ABO-TDF) are fundamental cryptographic primitives. And given the recent advances in quantum computing, it would be much desirable to develop new and improved lattice-based LTDF and ABO-TDF. In this work, we provide more compact constructions of LTDF and ABO-TDF based on the learning with errors (LWE) problem. In addition, our LWE-based ABO-TDF can allow smaller system parameters to support super-polynomially many injective branches in the construction of CCA secure public key encryption. As a core building tool, we provide a more compact homomorphic symmetric encryption schemes based on LWE, which might be of independent interest. To further optimize the ABO-TDF construction, we employ the full rank difference encoding technique. As a consequence, the results presented in this work can substantially improve the performance of all the previous LWE-based cryptographic constructions based upon LTDF and ABO-TDF.