There only exists one deterministic identity-based encryption (DIBE) scheme which is adaptively secure in the auxiliary-input setting, under the learning with errors (LWE) assumption. However, the master public key consists of O(λ) basic matrices. In this paper, we consider to construct adaptively secure DIBE schemes with more compact public parameters from the LWE problem. (i) On the one hand, we gave a generic DIBE construction from lattice-based programmable hash functions with high min-entropy. (ii) On the other hand, when instantiating our generic DIBE construction with four LPHFs with high min-entropy, we can get four adaptively secure DIBE schemes with more compact public parameters. In one of our DIBE schemes, the master public key only consists of ω(log⁡λ) basic matrices.

In this paper, we propose a hierarchical identity-based encryption (HIBE) scheme in the random oracle (RO) model based on the learning with rounding (LWR) problem over small modulus $q$. Compared with the previous HIBE schemes based on the learning with errors (LWE) problem, the ciphertext expansion ratio of our scheme can be decreased to 1/2. Then, we utilize the HIBE scheme to construct a deterministic hierarchical identity-based encryption (D-HIBE) scheme based on the LWR problem over small modulus. Finally, with the technique of binary tree encryption (BTE) we can construct HIBE and D-HIBE schemes in the standard model based on the LWR problem over small modulus.