This video covers the definitions for some basic algebraic structures, including groups and rings. I give examples of each and discuss how to verify the properties for each type of structure.

Views: 53125
James Hamblin

This video is useful for students of BSc/MSc Mathematics students. Also for students preparing IIT-JAM, GATE, CSIR-NET and other exams.

Views: 243528
Bhagwan Singh Vishwakarma

Learn the definition of a ring, one of the central objects in abstract algebra. We give several examples to illustrate this concept including matrices and polynomials.
If you’d like to help us make videos more quickly, you can support us on Patreon at
https://www.patreon.com/socratica
We also welcome Bitcoin donations! Our Bitcoin address is:
1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9
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**************
We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
http://amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
http://www.jmilne.org/math/CourseNotes/index.html
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Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison

Views: 130166
Socratica

Familiar algebraic systems: review and a look ahead.
GRF is an ALGEBRA course, and specifically a course about algebraic structures. This introductory
section revisits ideas met in the early part of Analysis I and in Linear Algebra I, to set the
scene and provide motivation.
https://people.maths.ox.ac.uk/flynn/genus2/sheets0405/grfnotes1011.pdf

Views: 857
Mysterious Mind

This video is useful for students of BTech/BE/Engineering/ BSc/MSc Mathematics students. Also for students preparing IIT-JAM, GATE, CSIR-NET and other exams.

Views: 554412
Bhagwan Singh Vishwakarma

Views: 6858
Internetwork Security

Rings are one of the key structures in Abstract Algebra. In this video we give lots of examples of rings: infinite rings, finite rings, commutative rings, noncommutative rings and more!
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Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison (@mlh496 on Twitter)
Produced by Kimberly Hatch Harrison

Views: 73149
Socratica

Cryptography and Network Security by Prof. D. Mukhopadhyay, Department of Computer Science and Engineering, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in

Views: 50870
nptelhrd

Coding Theory by Dr. Andrew Thangaraj, Department of Electronics & Communication Engineering, IIT Madras. For more details on NPTEL visit http://nptel.iitm.ac.in

Views: 26858
nptelhrd

We're in Dummit and Foote's 3rd edition, Part III

Views: 5526
James Cook

After the work of Diophantus, there was something of a lapse in interest in pure number theory for quite some while. Around 1300 Gersonides developed the connection between the Binomial theorem and combinatorics, and then in the 17th century the topic was again taken up, notably by Fermat, and then by Euler, Lagrange, Legendre and Gauss. We discuss several notable results of Fermat, including of course his famous last theorem, also his work on sums of squares, Pell's equation, primes, and rational points on curves. The rational parametrization of the Folium of Descartes is shown, using the technique of Fermat.
We also state Fermat's little theorem using the modular arithmetic language introduced by Gauss.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 32332
Insights into Mathematics

The prospect of outsourcing an increasing amount of data storage and management to cloud services raises many new privacy concerns that can be satisfactorily addressed if users encrypt the data they send to the cloud. If the encryption scheme is homomorphic, the cloud can still perform meaningful computations on the data, even though it is encrypted. In fact, we now know a number of constructions of fully homomorphic encryption schemes that allow arbitrary computation on encrypted data. In the last two years, solutions for fully homomorphic encryption have been proposed and improved upon, but all currently available options seem to be too inefficient to be used in practice. However, for many applications it is sufficient to implement somewhat homomorphic encryption schemes, which support a limited number of homomorphic operations. They can be much faster, and more compact than fully homomorphic schemes. This talk will focus on describing the recent somewhat homomor- phic encryption scheme of Brakerski and Vaikuntanathan, whose security relies on the ring learning with errors (RLWE) problem.

Views: 816
Microsoft Research

Additive and Multiplicative Inverse of elements in Galois Field.
link to my channel-
https://www.youtube.com/user/lalitkvashishtha
link to data structure and algorithm playlist -
https://www.youtube.com/watch?v=GbOW74e4xZE&list=PLLvKknWU7N4y_eGpQdg1Y-hORO7cxtoLU
link to information theory and coding techniques playlist -
https://www.youtube.com/watch?v=2qJ_mcjKYtk&list=PLLvKknWU7N4yDkIlN4YE-sXfFD4trDf6W
link to compiler design playlist -
https://www.youtube.com/watch?v=uAVkjTbB7Yc&list=PLLvKknWU7N4zpJWLqk7DXK26JwTB-gFmZ

Views: 33578
Lalit Vashishtha

Views: 14837
Laura Rickhoff

This video explores the Cryptography research group at the University of Bristol through an interview with the head of the group, Prof. Nigel Smart.

Views: 53
AzitaGhassemi Media

The Ring Learning-with-Errors problem, proposed by Lyubashevsky, Peikert and Regev in 2010, is a variant of the traditional Learning-with-Errors problem, and is an active research area in lattice based cryptography. It has drawn increased attention due to the important application to constructing homomorphic encryption schemes. The security of RLWE problems relies on the hardness of certain standard problems over ideal lattices. In the first part of the talk, I will review the basics of RLWE problems, the hardness proofs, and major RLWE encryption schemes. Then I will survey different attacks to RLWE, including our new attacks to non-dual RLWE in sub-cyclotomic fields and small error dual RLWE on prime cyclotomic fields. I will end by summarizing the security situation for various RLWE problems. This is joint work with Kristin Lauter and Katherine Stange.

Views: 2059
Microsoft Research

Learn and understand GF and various operations on elements using polynomial representation

Views: 17470
DrVikasThada

Learn the definition of a group - one of the most fundamental ideas from abstract algebra.
If you found this video helpful, please give it a "thumbs up" and share it with your friends!
To see more videos on Abstract Algebra, please watch our playlist:
https://www.youtube.com/watch?v=QudbrUcVPxk&list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6
If you’d like to help us make videos more quickly, you can support us on Patreon at
https://www.patreon.com/socratica
We also welcome Bitcoin donations! Our Bitcoin address is:
1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9
Thank you!!
**************
We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
http://amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
http://www.jmilne.org/math/CourseNotes/index.html
**************
Be sure to subscribe so you don't miss new lessons from Socratica:
http://bit.ly/1ixuu9W
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Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison

Views: 215304
Socratica

Winter School on Lattice-Based Cryptography and Applications, which took place at Bar-Ilan University between february 19 - 22.
The event's program: http://crypto.biu.ac.il/winterschool2012/
Dept. of Computer Science: http://www.cs.biu.ac.il/
Bar-Ilan University: http://www1.biu.ac.il/indexE.php

Views: 2927
barilanuniversity

What is RING THEORY? What does RING THEORY mean? RING THEORY meaning - RING THEORY definition - RING THEORY explanation.
Source: Wikipedia.org article, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license.
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.
Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to. For example, Hilbert's Nullstellensatz is a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra. Similarly, Fermat's last theorem is stated in terms of elementary arithmetic, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.
Noncommutative rings are quite different in flavour, since more unusual behavior can arise. While the theory has developed in its own right, a fairly recent trend has sought to parallel the commutative development by building the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on (non-existent) 'noncommutative spaces'. This trend started in the 1980s with the development of noncommutative geometry and with the discovery of quantum groups. It has led to a better understanding of noncommutative rings, especially noncommutative Noetherian rings. (Goodearl 1989)
For the definitions of a ring and basic concepts and their properties, see ring (mathematics). The definitions of terms used throughout ring theory may be found in the glossary of ring theory.

Views: 2440
The Audiopedia

Cyclic groups are the building blocks of abelian groups. There are finite and infinite cyclic groups. In this video we will define cyclic groups, give a list of all cyclic groups, talk about the name “cyclic,” and see why they are so essential in abstract algebra.
If you’d like to help us make videos more quickly, you can support us on Patreon at
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We also welcome Bitcoin donations! Our Bitcoin address is:
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**************
We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
http://amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
http://www.jmilne.org/math/CourseNotes/index.html
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Be sure to subscribe so you don't miss new lessons from Socratica:
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Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison

Views: 151163
Socratica

This video is useful for students of BTech/BE/Engineering/ BSc/MSc Mathematics students. Also for students preparing IIT-JAM, GATE, CSIR-NET and other exams.

Views: 515000
Bhagwan Singh Vishwakarma

Fermat's Little Theorem was observed by Fermat and proven by Euler, who generalized the theorem significantly. This theorem aids in dividing extremely large numbers and can aid in testing numbers to see if they are prime. For more advanced students, this theorem can be easily proven using basic group theory.
Prerequisites: To follow this video, you will want to first learn the basics of congruences.
If you found this video helpful, please share it with your friends!
You might like the other videos in our Number Theory Playlist:
https://www.youtube.com/watch?v=VLFjOP7iFI0&list=PLi01XoE8jYojnxiwwAPRqEH19rx_mtcV_
Don't forget to Subscribe to our channels so you'll hear about our newest videos:
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Subject: Number Theory
Teacher: Michael Harrison
Artist: Katrina de Dios

Views: 151861
Socratica

Visual Group Theory, Lecture 1.3: Groups in science, art, and mathematics
Groups are always lurking where symmetry arises. In this lecture, we explore many beautiful examples of groups that arise from natural symmetries in science, art, and mathematics. This includes shapes of molecules, repeating patterns in 1, 2, and 3 dimensions, and finally, how groups arise in braids.
Course webpage (with lecture notes, HW, etc.): http://www.math.clemson.edu/~macaule/math4120-online.html

Views: 12964
Professor Macauley

A solution to a typical exam question. See my other videos
https://www.youtube.com/channel/UCmtelDcX6c-xSTyX6btx0Cw/.

Views: 303457
Randell Heyman

It covers Euclid's Algorithm, Euclid's Algorithm: Tabular Method, Modular Arithmetic, Modular Arithmetic Operations, Modular Arithmetic Properties, Group, Cyclic Group, Ring, Field, Finite Fields or Galois Fields, Polynomial Arithmetic, Polynomial Arithmetic with Mod 2 Coefficients.

Views: 10002
Scholartica Channel

The notion of congruence modulo n is used to introduce the integers modulo n. Addition and multiplication are defined for the integers modulo n.

Views: 18302
learnifyable

B.Sc., B.C.A and other college courses
Arupsmath

Views: 15829
Arup Majumdar

• Counselling Guruji is our latest product & a well-structured program that answers all your queries related to Career/GATE/NET/PSU’s/Private Sector etc.
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• Link for the complete playlist of Discrete Mathematics is:
Relations:https://www.youtube.com/playlist?list=PLmXKhU9FNesTpQNP_OpXN7WaPwGx7NWsq
Graph Theory: https://www.youtube.com/playlist?list=PLmXKhU9FNesS7GpOddHDX3ZCl86_cwcIn
Group Theory: https://www.youtube.com/playlist?list=PLmXKhU9FNesQrSgLxm6zx3XxH_M_8n3LA
Proposition:https://www.youtube.com/playlist?list=PLmXKhU9FNesQxcibunbD82NTQMBKVUO1S
Set Theory: https://www.youtube.com/playlist?list=PLmXKhU9FNesTSqP8hWDncxpCj8a4uzmu7
• Links for the books that we recommend for Discrete Mathematics are:
1. Discrete Mathematics and Its Applications (Writer: Kenneth Rosen) (Publisher: McGraw Hill Education)
https://amzn.to/2NV9viK
2. Graph Theory with Applications to Engineering and Computer Science (Writer: Deo Narsingh) (Publisher: Phi)
https://amzn.to/2NSiwcc
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• Links for Hindi playlists of other Subjects are:
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TOC: https://www.youtube.com/playlist?list=PLmXKhU9FNesSdCsn6YQqu9DmXRMsYdZ2T
OS: https://www.youtube.com/playlist?list=PLmXKhU9FNesSFvj6gASuWmQd23Ul5omtD
Digital Electronics: https://www.youtube.com/playlist?list=PLmXKhU9FNesSfX1PVt4VGm-wbIKfemUWK
Data Structures: https://www.youtube.com/playlist?list=PLmXKhU9FNesRRy20Hjr2GuQ7Y6wevfsc5
Computer Networks: https://www.youtube.com/playlist?list=PLmXKhU9FNesSjFbXSZGF8JF_4LVwwofCd
Algorithm: https://www.youtube.com/playlist?list=PLmXKhU9FNesQJ3rpOAFE6RTm-2u2diwKn
• About this video:
This video discusses about the basics of closure property and also about the algebraic structures in discrete mathematics. This video discusses the basic definition of algebraic properties along with some examples to clear your understanding on algebraic structures.
Notes:
● Closure Property: A set A with respect to a operator * is said to satisfy the closure property
if, ∀ a, b ∈ A
then, a * b ∈ A
● Algebraic Structure: A set ‘A’ w.r.t operator ‘*’ satisfy closure property then it is called algebraic structure.
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Views: 131600
KNOWLEDGE GATE

During the IBM Research 5 in 5 Science Slam at IBM Think 2018, IBM researcher Cecilia Boschini explains one of the technologies that will change the world in the next five years: lattice cryptography. Learn more at http://ibm.biz/five-in-five.

Views: 4331
IBM Research

This was for the MAO Math Presentation Competition. I won! :D

Views: 32647
Riverninj4

Benoît Libert and San Ling and Fabrice Mouhartem and Khoa Nguyen and Huaxiong Wang. Talk at Asiacrypt 2016. See http://www.iacr.org/cryptodb/data/paper.php?pubkey=27877

Views: 115
TheIACR

Introduction
Ensembles + Axiomes + Règles + Lois = Structure Algébrique
Structures + extra Axiomes = Groupes Anneaux et Corps
Problème des racines des équations polynomiales de degré supérieure a 4.

Views: 502
Lifenote School

Winter School on Lattice-Based Cryptography and Applications, which took place at Bar-Ilan University between february 19 - 22.
The event's program: http://crypto.biu.ac.il/winterschool2012/
Dept. of Computer Science: http://www.cs.biu.ac.il/
Bar-Ilan University: http://www1.biu.ac.il/indexE.php

Views: 7588
barilanuniversity

Vadim Lyubashevsky's August 13, 2013 lecture at the UCI Workshop on Lattices with Symmetry. The last 10 minutes of audio are missing.

Views: 1445
Workshop on Lattices with Symmetry

Winter School on Lattice-Based Cryptography and Applications, which took place at Bar-Ilan University between february 19 - 22.
The event's program: http://crypto.biu.ac.il/winterschool2012/
Dept. of Computer Science: http://www.cs.biu.ac.il/
Bar-Ilan University: http://www1.biu.ac.il/indexE.php

Views: 3793
barilanuniversity

Algebra is my favorite branch of mathematics. In this video, I try to describe what the subject is about, assuming as little mathematical background as possible, and within 10 minutes.

Views: 2378
Hsing Liu

Views: 2872
barilanuniversity

Part 2 of lecture 7 from my ring theory lecture playlist. Topics discussed include properties of rings of fractions and modules of fractions.

Views: 146
For Your Math

Jeff Hoffstein's August 31 presentation on "Somewhat Homomorphic Encryption via Number Fields and Finite Fields" at the 2015 UCI Mathematics of Cryptography Conference

Views: 266
Conference on Mathematics of Cryptography

• Counselling Guruji is our latest product & a well-structured program that answers all your queries related to Career/GATE/NET/PSU’s/Private Sector etc.
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• Link for the complete playlist of Discrete Mathematics is:
Relations:https://www.youtube.com/playlist?list=PLmXKhU9FNesTpQNP_OpXN7WaPwGx7NWsq
Graph Theory: https://www.youtube.com/playlist?list=PLmXKhU9FNesS7GpOddHDX3ZCl86_cwcIn
Group Theory: https://www.youtube.com/playlist?list=PLmXKhU9FNesQrSgLxm6zx3XxH_M_8n3LA
Proposition:https://www.youtube.com/playlist?list=PLmXKhU9FNesQxcibunbD82NTQMBKVUO1S
Set Theory: https://www.youtube.com/playlist?list=PLmXKhU9FNesTSqP8hWDncxpCj8a4uzmu7
• Links for the books that we recommend for Discrete Mathematics are:
1. Discrete Mathematics and Its Applications (Writer: Kenneth Rosen) (Publisher: McGraw Hill Education)
https://amzn.to/2NV9viK
2. Graph Theory with Applications to Engineering and Computer Science (Writer: Deo Narsingh) (Publisher: Phi)
https://amzn.to/2NSiwcc
• Check out our website http://www.knowledgegate.in/
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• Links for Hindi playlists of other Subjects are:
DBMS: https://www.youtube.com/playlist?list=PLmXKhU9FNesR1rSES7oLdJaNFgmuj0SYV
TOC: https://www.youtube.com/playlist?list=PLmXKhU9FNesSdCsn6YQqu9DmXRMsYdZ2T
OS: https://www.youtube.com/playlist?list=PLmXKhU9FNesSFvj6gASuWmQd23Ul5omtD
Digital Electronics: https://www.youtube.com/playlist?list=PLmXKhU9FNesSfX1PVt4VGm-wbIKfemUWK
Data Structures: https://www.youtube.com/playlist?list=PLmXKhU9FNesRRy20Hjr2GuQ7Y6wevfsc5
Computer Networks: https://www.youtube.com/playlist?list=PLmXKhU9FNesSjFbXSZGF8JF_4LVwwofCd
Algorithm: https://www.youtube.com/playlist?list=PLmXKhU9FNesQJ3rpOAFE6RTm-2u2diwKn
• About this video:
This video discusses the practice problems on algebraic structures, semi-group and monoids. You’ll be learning about how to solve these common problems asked frequently in GATE exam.
Notes:
Practice Problem:
Let A = { 1,2,3,4,…….∞ } and a binary operation ‘+’ is defined by a + b = ab ∀ a,b ∈ A. Which of the following is true ?
A. ( A, + ) is a semi group but not monoid
B. ( A, + ) is a monoid but not group
C. ( A, + ) is a group
D. ( A, + ) is not a semi group
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Views: 60628
KNOWLEDGE GATE

Vadim Lyubashevsky. Talk at Asiacrypt 2016. See http://www.iacr.org/cryptodb/data/paper.php?pubkey=27914

Views: 163
TheIACR

Use of Galois Binary Fields in cryptography, and the Advanced Encryption Standard (AES) encryption scheme. Corresponding notes:
https://www.cs.uaf.edu/2015/spring/cs463/lecture/03_23_AES.html
Course lecture for CS 463/680, Cryptography and Data Security
https://www.cs.uaf.edu/courses/cs463/2015-spring/

Views: 126
Orion Lawlor

Mayank Rathee demos a Python implementation of the homomorphic encryption scheme by Yoshinori Aono et al. for the OpenMined ecosystem. For the sake of demo, we have pruned down the uniform sampling space to a smaller subset of the otherwise larger space. This is why the ciphertext components look smaller in comparison to the modulus Q.

Views: 1480
OpenMined