At PQCrypto 2016, Szepieniec et al. proposed a new type of trapdoor called Extension Field Cancellation (EFC) for constructing secure multivariate encryption cryptosystems. They also specifically suggested two schemes EFC−p and EFC−pt2 that apply this trapdoor and some modifiers. Although both of them seem to avoid all attacks used for cryptanalysis on multivariate cryptography, their decryption efficiency has room for improvement. On the other hand, their security was analyzed mainly through an algebraic attack of computing the Gröbner basis of the public key, and there possibly exists more effective attacks. In this paper, we introduce a more efficient decryption approach for EFC−p and EFC−pt2, which manages to avoid all redundant computation involved in the original decryption algorithms without altering their public key. In addition, we estimate the secure parameters for EFC−p and EFC−pt2 through a hybrid attack of algebraic attack and exhaustive search.