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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 have designed three fast implementations of a recently proposed family of hash functions Edon–$ℛ$. They produce message digests of length $n=256,384,512$ bits and project security of $2^{\frac{n}{2}}$ hash computations for finding collisions and $2^n$ hash computations for finding preimages and second preimages. The design is not the classical Merkle-Damgård but can be seen as wide-pipe iterated compression function. Moreover the design is based on using huge quasigroups of orders $2^{256}$, $2^{384}$ and $2^{512}$ that are constructed by using only bitwise...