This thesis explores a post-quantum encryption scheme based on the multivariate quadratic problem. The original scheme has been introduced by Esposito et al.[EFR24]. This thesis proposes a modified version where encryption has been made deterministic and a permutation map has been included in order to increase security against the best known attack, so as to reduce the size of the security parameters. Crypto analysis has been performed A new security assumption has been introduced and se- curity has been proved based on that assumption. A KEM and a multi- encryption construction have been proposed. A C implementation of the system has been realized.