Let $q$ be a positive integer. An algebra is said to have the property $(2,q)$ if all of its subalgebras generated by two distinct elements have exactly $q$ elements. A variety $𝒱$ of algebras is a variety with the property $(2,q)$ if every member of $𝒱$ has the property $(2,q)$. Such varieties exist only in the case of $q$ prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property $(2,q)$, $2<q$, blocks of Steiner system of type $(2,q)$ are obtained. The stated correspondence between Steiner...

We give a short account of the construction and properties of left neofields. Most useful in practice seem to be neofields based on the cyclic group and particularly those having an additional divisibility property, called D-neofields. We shall give examples of applications to the construction of orthogonal latin squares, to the design of tournaments balanced for residual effects and to cryptography.

AMS Subj. Classification: Primary 20N05, Secondary 94A60The intention of this research is to justify deployment of quasigroups in cryptography, especially with new quasigroup based cryptographic hash function NaSHA as a runner in the First round of the ongoing NIST SHA-3 competition. We present new method for fast generation of huge quasigroup operations, based on the so-called extended Feistel networks and modification of the Sade’s diagonal method. We give new design of quasigroup based family of...

Automorphic loops are loops in which all inner mappings are automorphisms. We study a generalization of the dihedral construction for groups. Namely, if $(G,+)$ is an abelian group, $m\ge 1$ and $\alpha \in Aut\left(G\right)$, let $Dih(m,G,\alpha )$ be defined on ${\mathbb{Z}}_m\times G$ by $$(i,u)(j,v)=(i\oplus j,({(-1)}^{j}u+v){\alpha }^{ij}).$$ The resulting loop is automorphic if and only if $m=2$ or (${\alpha }^2=1$ and $m$ is even). The case $m=2$ was introduced by Kinyon, Kunen, Phillips, and Vojtěchovský. We present several structural results about the automorphic dihedral loops in both cases.

This note contains Sylow's theorem, Lagrange's theorem and Hall's theorem for finite Bruck loops. Moreover, we explore the subloop structure of finite Bruck loops.

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...