The Quasigroup Block Cipher and its Analysis

This thesis discusses the Quasigroup Block Cipher (QGBC) and its analysis. We first present the basic form of the QGBC and then follow with improvements in memory consumption and security. As a means of analyzing the system, we utilize tools such as the NIST Statistical Test Suite, auto and crosscorrelation, then linear and algebraic cryptanalysis. Finally, as we review the results of these analyses, we propose improvements and suggest an algorithm suitable for low-cost FPGA implementation

NaSHA

We propose the NaSHA-(m, k, r) family of cryptographic hash functions, based on quasigroup transformations. We use huge quasigroups defined by extended Feistel networks from small bijections and a novel design principle: the quasigroup used in every iteration of the compression function is different and depends on the processed message block. We present in all details of the implementations of NaSHA-(m, 2, 6) where m in {224, 256, 384, 512}

Abstracts of Ph.D. theses in mathematics

summary:Leischner, Pavel: Spatial imagination development of the secondary school pupils. Mašíček, Libor: Diagnostics and sensitivity of robust models. Duintjer Tebbens, Erik Jurjen: Modern methods for solving linear problems. Matonoha, Ctirad: Numerical realization of trust region methods. Duda, Jakub: Delta convexity, metric projection and negligible sets. Smrčka, Michael: Choquet's theory in function spaces. Hanika, Jiří: Search problems and bounded arithmetic. Pawlas, Zbyněk: Asymptotics in stochastic geometry. Bodlák, Karel: Methods of stereology and spatial statistics in applications. Čapek, Václav: M-smoothers Zvára, Petr: Prediction in non-linear autoregressive processes. Blanda, Jiří: Pricing of life insurance products Finfrle, Pavel: Model for calculation of liability value arising from life insurance. Finěk Václav: Orthonormal wavelets and their applications. Stanovský David : Left distributive left quasigroups. Koblížková, Michaela: Polyhedra and secondary school mathematics. Krýsl, Svatopluk: Invariant differential operators for projective contact geometries. Šmíd, Dalibor: Properties of invariant differential operators. Šmíd, Martin: Notes on approximation of stochastic programming problems. Komárková, Lenka: Change point problem for censored data. Kechlibar, Marian: Commutative algebra and cryptography

On free quasigroups and quasigroup representations

This work consists of three parts. The discussion begins with \emph{linear quasigroups}. For a unital ring $S$, an $S$-linear quasigroup is a unital $S$-module, with automorphisms $\rho$ and $\lambda$ giving a (nonassociative) multiplication $x\cdot y=x^\rho+y^\lambda$. If $S$ is the field of complex numbers, then ordinary characters provide a complete linear isomorphism invariant for finite-dimensional $S$-linear quasigroups. Over other rings, it is an open problem to determine tractably computable isomorphism invariants. The paper investigates this isomorphism problem for $\mathbb{Z}$-linear quasigroups. We consider the extent to which ordinary characters classify $\mathbb{Z}$-linear quasigroups and their representations of the free group on two generators. We exhibit non-isomorphic $\mathbb{Z}$-linear quasigroups with the same ordinary character. For a subclass of $\mathbb{Z}$-linear quasigroups, equivalences of the corresponding ordinary representations are realized by permutational intertwinings. This leads to a new equivalence relation on $\mathbb{Z}$-linear quasigroups, namely permutational similarity. Like the earlier concept of central isotopy, permutational similarity is intermediate between isomorphism and isotopy. The story progresses with a representation of the free quasigroup on a single generator. This provides the motivation behind the study of \emph{peri-Catalan numbers}. While Catalan numbers index the number of length $n$ magma words in a single generator, peri-Catalan numbers index the number of length $n$ reduced form quasigroup words in a single generator. We derive a recursive formula for the $n$-th peri-Catalan number. This is a new sequence in that it is not on the Online Encyclopedia of Integer Sequences

Some Relations on Paratopisms and An Intuitive Interpretation on the Adjugates of a Latin Square

This paper will present some intuitive interpretation of the adjugate transformations of arbitrary Latin square. With this trick, we can generate the adjugates of arbitrary Latin square directly from the original one without generating the orthogonal array. The relations of isotopisms and adjugate transformations in composition will also be shown. It will solve the problem that when F1*I1=I2*F2 how can we obtain I2 and F2 from I1 and F1, where I1 and I2 are isotopisms while F1 and F2 are adjugate transformations and * is the composition. These methods could distinctly simplify the computation on a computer for the issues related to main classes of Latin squares.Comment: Any comments and criticise are appreciate