Some Results on the Upper Convex Densities of the Self-Similar Sets at the Contracting-Similarity Fixed Points

In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered.

Lattice-based hierarchical identity-based broadcast encryption scheme in the standard model

Lattice-based hierarchical identity-based broadcast encryption(H-IBBE)schemes have broad application prospects in the quantum era,because it reduces the burden of private key generator(PKG)and is suitable for one-to-many communication.However,previous lattice-based H-IBBE schemes are mostly constructed in the random oracle model with more complex trapdoor delegation process and have lower practical application.A lattice-based H-IBBE is proposed in the fixed dimension under the standard model,which mainly consists of binary tree encryption(BTE)system,MP12 trapdoor function and ABB10b trapdoor delegation algorithm.First,this paper uses BTE system to eliminate the random oracle so that the scheme can be implemented under the standard model,and it also uses MP12 trapdoor function to reduce trapdoor generation complexity and obtains a safe and efficient trapdoor matrix;Second,this paper uses ABB10b trapdoor delegation algorithm to delegate user爷s private key,and the trapdoor matrices'dimensions are the same before and after the trapdoor delegation.Comparative analysis shows that trapdoor delegation process reduces complexity,and the size of cipher-text and trapdoor matrix does not increase with deeper trapdoor delegation process.This paper achieves indistinguishability of cipher-texts under a selective chosen-cipher-text and chosen-identity attack(INDr-sID-CCA)security in the standard model based on learning with errors(LWE)hard assumption.