The quantum algorithm threat on the security of public key cryptography pushes people to initiate the research of post-quantum cryptography represented by lattice cryptography. Efficient lattice cryptography is based on the parameter range that dissatisfies the proven security condition, whose attack cost model depends on heuristics and experimental results, leading to a severe impact on concrete security estimation and parameter selection of lattice cryptography. This study firstly recalls some preliminary mathematics definitions and propositions of lattice cryptography and lattice hard problems, including solving algorithms of the latter. Then, cost models of these algorithms are reviewed and are partitioned into 2008 and 2016 cost models. Applications on a special type of lattice is also reviewed. This review discusses concrete security analysis of two types of lattice cryptography which are based on two distinct hard problems, which offers a criteria to parameter selection of lattice cryptography.