Data stored in the cloud servers, keyword search, and access controls are two important capabilities which should be supported. Public-keyword encryption with keyword search (PEKS) and attribute based encryption (ABE) are corresponding solutions. Meanwhile, as we step into postquantum era, pairing related assumption is fragile. Lattice is an ideal choice for building secure encryption scheme against quantum attack. Based on this, we propose the first mathematical model for lattice-based authorized searchable encryption. Data owners can sort the ciphertext by specific keywords such as time; data users satisfying the access control hand the trapdoor generated with the keyword to the cloud sever; the cloud sever sends back the corresponding ciphertext. The security of our schemes is based on the worst-case hardness on lattices, called learning with errors (LWE) assumption. In addition, our scheme achieves attribute-hiding, which could protect the sensitive information of data user.

<abstract> <p>In this paper, we study the initial-boundary value problem for a class of fractional $ p $-Laplacian Kirchhoff diffusion equation with logarithmic nonlinearity. For both subcritical and critical states, by means of the Galerkin approximations, the potential well theory and the Nehari manifold, we prove the global existence and finite time blow-up of the weak solutions. Further, we give the growth rate of the weak solutions and study ground-state solution of the corresponding steady-state problem.</p> </abstract>

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