Middle-product learning with errors \(({\text {MP}}-{\text {LWE}})\) introduced by Rosca et al. (CRYPTO 2017) is a variant of polynomialLearning with errors learning with errors \(({\text {Poly}}-{\text {LWE}})\) . \({\text {MP}}-{\text {LWE}}\) -based cryptosystems offer higher efficiency than \(\text {LWE}\) -based cryptosystems. Moreover, a reduction from the \({\text {Poly}}-{\text {LWE}}\) problems to the \({\text {MP}}-{\text {LWE}}\) problem exists, the \({\text {MP}}-{\text {LWE}}\) problem is at least as hard as the \({\text {Poly}}-{\text {LWE}}\) problems. After the seminal work by Rosca et al., \({\text {MP}}-{\text {LWE}}\) -based cryptography has progressed. In particular, the \({\text {MP}}-{\text {LWE}}\) -based public key encryption scheme called Titanium was submitted in the NIST post-quantum cryptography competition process. Moreover, the trapdoor technique for \({\text {MP}}-{\text {LWE}}\) developed, identity-based encryption schemes were proposed. In this paper, we give a survey on \({\text {MP}}-{\text {LWE}}\) cryptography.