On a special class of multivariate quadratic quasigroups (MQQs)

Abstract

In this paper, we study a special class of recently introduced quasigroups called multivariate quadratic quasigroups (MQQs) and solve several open research problems about them. Our main contributions are threefold. The first is to provide a standard form of the MQQ generating function. Secondly, we show how to explicitly construct MQQs of higher orders and give lower bounds on the number of MQQs, which so far were open problems. Last, but not least, we refine the definition of MQQs of different types by introducing the notion of “MQQs of strict type”. The new concept has the advantage of being invariant under invertible affine transformations over the set of the rows, columns and symbols of the multiplication table of an MQQ. It is therefore better suited to characterize the complexity of the underneath multivariate quadratic system.

Keywords

• quasigroup
• multivariate quadratic system
• boolean function
• matrix algebra