© The Institution of Engineering and Technology 2019 Multivariate public key cryptography is one of the main candidates for post-quantum cryptography. Rainbow, an improved (multi-layer) version of unbalanced oil and vinegar (UOV), is one of the most famous multivariate signature schemes that is a promising candidate for NIST standardisation. At INDOCRYPT 2017, Beullens and Preneel introduced a new variant LUOV of UOV. Their idea is to generate a UOV scheme over the binary field L = F2 and then lift it into a bigger field K = F2r and hence dramatically reduce the public key size. In this study, the authors first theoretically deduce the choice for the subfield L (which is different from F2) which results in smaller signature sizes (up to 40%). Moreover, they extend the idea to Rainbow and theoretically yield the optimal choice for the subfield L over which a Rainbow is generated before being lifted to K. As a result, they can reduce the public key size of the obtained Rainbow scheme up to at least 36%.