Certificateless public key encryption solves the problems of establishing public-key infrastructure for traditional public key encryption and resolving key escrow for identity-based encryption. Equality test is an extremely useful property that enables the ability of checking whether two ciphertexts encrypting the same message. Qu et al. (Information Science 2019) introduced the notion of certificateless public key encryption with equality test, together with four types of adversaries, that solves certificate manangement and key escrow problems of public key encryption with equality test and identity-based encryption with equality test, and proposed a first scheme based on Bilinear Diffie-Hellman assumption in random oracle model. In this paper, we propose the first lattice-based in standard model whose security is reduced to the hardness of the learning with errors problem. In particular, we prove that our schemes are secure against two types of selective-identity adversaries introduced by Qu et al.