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 $q_{\setminus }$ defined by: $q_{\setminus }(u,v)=w\iff q(u,w)=v$ that also defines a left multivariate quasigroup. However, in general, $(Q,q_{\setminus })$ 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...

We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences ${\sum }_{i=1}^k{a}_i{x}_i\equiv b(modn)$. In particular, we obtain explicit expressions for the number of solutions, where $x_i$’s are squares modulo $n$. In addition, we obtain expressions for the number of solutions with order restrictions $x_1\ge \cdots \ge x_k$ or with strict order restrictions $x_1>\cdots >x_k$ in some special cases. In these results, the expressions for the number of solutions involve Ramanujan...