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Edon- ℛ ( 256 , 384 , 512 ) – an efficient implementation of Edon- ℛ family of cryptographic hash functions

Danilo Gligoroski, Svein Johan Knapskog (2008)

Commentationes Mathematicae Universitatis Carolinae

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 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...

Embedding 3 -homogeneous latin trades into abelian 2 -groups

Nicholas J. Cavenagh (2004)

Commentationes Mathematicae Universitatis Carolinae

Let T be a partial latin square and L be a latin square with T ⊆ L . We say that T is a latin trade if there exists a partial latin square T ' with T ' ∩ T = ∅ such that ( L ∖ T ) ∪ T ' 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.

Enumeration of nilpotent loops up to isotopy

Lucien Clavier (2012)

Commentationes Mathematicae Universitatis Carolinae

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 2 q up to isotopy, instead of isomorphy.

Enveloping algebras of Malcev algebras

Murray R. Bremner, Irvin R. Hentzel, Luiz A. Peresi, Marina V. Tvalavadze, Hamid Usefi (2010)

Commentationes Mathematicae Universitatis Carolinae

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...

Expansion in finite simple groups of Lie type

Emmanuel Breuillard, Ben J. Green, Robert Guralnick, Terence Tao (2015)

Journal of the European Mathematical Society

We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].

Extending the structural homomorphism of LCC loops

Piroska Csörgö (2005)

Commentationes Mathematicae Universitatis Carolinae

A loop Q is said to be left conjugacy closed if the set A = { L x / x ∈ 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 ( Q ) be its inner mapping group, M ( Q ) its multiplication group. By Drápal’s theorem [3, Theorem 2.8] there exists a homomorphism Λ : ℒ → I ( Q ) determined by L x → R x - 1 L x . In this short note we examine different possible extensions of this Λ and the uniqueness of these extensions.

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