Check for On the Construction of Associative and Commutative Bilinear Multivariate Quadratic Quasigroups of Order 2n

Abstract

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" can be represented as vector valued Boolean functions from {0, 1}"×{0, 1}" to {0, 1}". 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.