On the Construction of Associative and Commutative Bilinear Multivariate Quadratic Quasigroups of Order \(2^n\)

Quasigroups have various applications in mathematics, computer science, and cryptography. In coding theory and cryptography they have been used in error-correcting codes, error-detection codes, to construct key exchange protocols and cryptographic primitives. There are also used in graph theory, experimental design and combinatorial designs. Quasigroups of order \(2^n\) can be represented as vector valued Boolean functions from \(\{0,1\}^n \times \{0,1\}^n\) to \(\{0,1\}^n\) . When the order of each coordinate functions is at most two, they are called Multivariate Quadratic Quasigroups (MQQ). In this paper we give a description of the functions representing Bilinear MQQ quasigroups, with a special focus on quasigroups that are commutative or associative.