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left multivariate quadratic quasigroup; left parastrophe; algebraic degree; matrix of Boolean polynomials
A left quasigroup $(Q,q)$ of order $2^w$ that can be represented as a vector of Boolean functions of degree 2 is called a left multivariate quadratic quasigroup (LMQQ). For a given LMQQ there exists a left parastrophe operation $q_{\backslash}$ defined by: $q_{\backslash}(u,v)= w \Leftrightarrow q(u,w)= v$ that also defines a left multivariate quasigroup. However, in general, $(Q,q_{\backslash})$ is not quadratic. Even more, representing it in a symbolic form may require exponential time and space. In this work we investigate the problem of finding a subclass of LMQQs whose left parastrophe is also quadratic (i.e. is also an LMQQ), and in the same time can be easily constructed. These LMQQs are affine in the second argument, and their left parastrophe can be easily expressed from the quasigroup operation. We give necessary and sufficient conditions for an LMQQ of this type to have a left parastrophe that is also an LMQQ. Based on this, we distinguish a special class that satisfies our requirements and whose construction is deterministic and straightforward.
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