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...

Let $T$ be a partial latin square and $L$ be a latin square with $T\subseteq L$. We say that $T$ is a latin trade if there exists a partial latin square $T^{\text{'}}$ with $T^{\text{'}}\cap T=\varnothing$ such that $(L\setminus T)\cup T^{\text{'}}$ is a latin square. A $k$-homogeneous latin trade is one which intersects each row, each column and each entry either $0$ or $k$ times. In this paper, we show the existence of $3$-homogeneous latin trades in abelian $2$-groups.

We modify tools introduced in [Daly D., Vojtěchovský P., Enumeration of nilpotent loops via cohomology, J. Algebra 322 (2009), no. 11, 4080–4098] to count, for any odd prime $q$, the number of nilpotent loops of order $2q$ up to isotopy, instead of isomorphy.

We first discuss the construction by Pérez-Izquierdo and Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras (both nonassociative and alternative) of the 4-dimensional solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We include a proof (due to Shestakov) that the universal alternative enveloping algebra of the real 7-dimensional simple Malcev algebra...

A loop $Q$ is said to be left conjugacy closed if the set $A=\{L_x/x\in Q\}$ is closed under conjugation. Let $Q$ be an LCC loop, let $ℒ$ and $ℛ$ be the left and right multiplication groups of $Q$ respectively, and let $I\left(Q\right)$ be its inner mapping group, $M\left(Q\right)$ its multiplication group. By Drápal’s theorem [3, Theorem 2.8] there exists a homomorphism $\Lambda :ℒ\to I\left(Q\right)$ determined by $L_x\to R_x^{-1}{L}_x$. In this short note we examine different possible extensions of this $\Lambda$ and the uniqueness of these extensions.