An Introduction to Quasigroups and Their Representations

Let O K be a discrete valuation ring of mixed characteristics p0, pq, with residue field k. Using work of Sekiguchi and Suwa, we construct some finite flat O K -models of the group scheme µ p n ,K of p n -th roots of unity, which we call Kummer group schemes. We carefully set out the general framework and algebraic properties of this construction. When k is perfect and O K is a complete totally ramified extension of the ring of Witt vectors W pkq, we provide a parallel study of the Breuil-Kisin modules of finite flat models of µ p n ,K , in such a way that the construction of Kummer groups and Breuil-Kisin modules can be compared. We compute these objects for n ď 3. This leads us to conjecture that all finite flat models of µ p n ,K are Kummer group schemes.

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“…-Quasigroups and loops. We start with some definitions from quasigroup theory, referring to the book of Smith [28] for more details.…”

Section: The Loop Of µ-Matricesmentioning

confidence: 99%

Models of group schemes of roots of unity

Mézard¹,

Romagny²,

Tossici³

2013

Ann. inst. Fourier

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“…Vector matrix representation of any octonion can be denoted as vector matrices according to following vector matrix representations [9]. 2) In order to prevent confusion we will use di¤erent notations for both octonions and their representations.…”

Section: Vector Matrix Representation Of Octonion Algebramentioning

confidence: 99%

Vector matrix representation of octonions and their geometry

HALICI

2018

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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“…Since it is based on quasigroups we give several definitions about quasigroups. Additional information about quasigroups reader can find in [2,3,15,23].…”

Section: Preliminariesmentioning

confidence: 99%

High Performance Implementation of a Public Key Block Cipher - MQQ, for FPGA Platforms

El-Hadedy

Gligoroski

Knapskog

2008

2008 International Conference on Reconfigurable Computing and FPGAs

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-We have implemented in FPGA recently published class of public key algorithms -MQQ, that are based on quasigroup string transformations. Our implementation achieves decryption throughput of 399 Mbps on an Xilinx Virtex-5 FPGA that is running on 249.4 MHz. The encryption throughput of our implementation achieves 44.27 Gbps on four Xilinx Virtex-5 chips that are running on 276.7 MHz. Compared to RSA implementation on the same FPGA platform this implementation of MQQ is 10,000 times faster in decryption, and is more than 17,000 times faster in encryption.

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An Introduction to Quasigroups and Their Representations

Cited by 95 publications

References 0 publications

Models of group schemes of roots of unity

Models of group schemes of roots of unity

Vector matrix representation of octonions and their geometry

High Performance Implementation of a Public Key Block Cipher - MQQ, for FPGA Platforms

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