An Algorithm for Judging and Generating Bilinear Multivariate Quadratic Quasigroups

Multivariate Quadratic Quasigroups (MQQs) as a suitable class of quasigroups for multivariate public key cryptography recently have been an important mathematical tool in information security field. In this paper, we propose a necessary and sufficient condition to verify whether a quasigroup given by its multiplication table is a bilinear MQQ, which shows that checking whether an arbitrary quasigroup is a bilinear MQQ is equivalent to solving a simple matrix equation. Based on this condition, a deterministic algorithm is proposed to judge whether a given quasigroup is a bilinear MQQ and then obtain the corresponding Boolean functions if it is. An example is given to show the validity of our results.