Number theoretic algorithms

Extended Euclidean algorithm

In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs.It allows one to compute also, with almost no extra cost, the quotients of a and b by their greatest common divisor. Extended Euclidean algorithm also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials. The extended Euclidean algorithm is particularly useful when a and b are coprime. With that provision, x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Similarly, the polynomial extended Euclidean algorithm allows one to compute the multiplicative inverse in algebraic field extensions and, in particular in finite fields of non prime order. It follows that both extended Euclidean algorithms are widely used in cryptography. In particular, the computation of the modular multiplicative inverse is an essential step in the derivation of key-pairs in the RSA public-key encryption method. (Wikipedia).

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The extended Euclidean algorithm in one simple idea

An intuitive explanation of the extended Euclidean algorithm as a simple modification of the Euclidean algorithm. This video is part of playlist on GCDs and the Euclidean algorithm: https://www.youtube.com/playlist?list=PLrm9Y---qlNxXccpwYQfllCrHRJWwMky-

From playlist GCDs and Euclidean algorithm

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Compute E Solution - Applied Cryptography

This video is part of an online course, Applied Cryptography. Check out the course here: https://www.udacity.com/course/cs387.

From playlist Applied Cryptography

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Extended Fundamental Theorem of Calculus

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Extended Fundamental Theorem of Calculus. You can use this instead of the First Fundamental Theorem of Calculus and the Second Fundamental Theorem of Calculus. - Formula - Proof sketch of the formula - Six Examples

From playlist Calculus

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Extended Euclidean Algorithm to Solve Linear Diophantine Equation

#shorts #mathonshorts Check out the videos on the GCD, Euclidean algorithms here Two Basic Theorems on gcd (Greatest Common Divisors) of Two Integers (Bezout's Identity) https://youtu.be/mQMksLNscY4 An Example of GCD, and Extended Euclidean Algorithm In Finding the Bezout Coefficients h

From playlist Elementary Number Theory

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12_1_1 Introduction to Taylor Polynomials

An introduction to expand a function into a Taylor polynomial.

From playlist Advanced Calculus / Multivariable Calculus

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Extended Euclidean Algorithm

This video walks through the technique for finding the GCD of two integers (not both zero), d = gcd(m,n), and then finding coefficients a and b for which d = am + bn.

From playlist Math

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12_2_1 Taylor Polynomials of Multivariable Functions

Now we expand the creation of a Taylor Polynomial to multivariable functions.

From playlist Advanced Calculus / Multivariable Calculus

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Extended Euclidean Algorithm

Gary Rubinstein teaches how to do the substitution method of the extended euclidean algorithm.

From playlist Cryptography, Security

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Number Theory | Extended Euclidean Algorithm Example 2

We use the extended Euclidean algorithm to write the greatest common divisor of two natural numbers as a linear combination of them.

From playlist Divisibility and the Euclidean Algorithm

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A Short Course in Algebra and Number Theory - Elementary Number Theory

To supplement a course taught at The University of Queensland's School of Mathematics and Physics I present a very brief summary of algebra and number theory for those students who need to quickly refresh that material or fill in some gaps in their understanding. This is the fourth lectu

From playlist A Short Course in Algebra and Number Theory

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2.1.4 Pulverizer: Video

MIT 6.042J Mathematics for Computer Science, Spring 2015 View the complete course: http://ocw.mit.edu/6-042JS15 Instructor: Albert R. Meyer License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 6.042J Mathematics for Computer Science, Spring 2015

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Math Major Examples — Number Theory #3

⭐Support the channel⭐ Patreon: https://www.patreon.com/michaelpennmath Merch: https://teespring.com/stores/michael-penn-math My amazon shop: https://www.amazon.com/shop/michaelpenn ⭐my other channels⭐ Main Channel: https://www.youtube.com/michaelpennmath non-math podcast: http

From playlist Number Theory

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Number Theory | Extended Euclidean Algorithm Example #1

We use the extended Euclidean algorithm to write the greatest common divisor of two natural numbers as a linear combination of them.

From playlist Divisibility and the Euclidean Algorithm

Related pages

Quotient ring | Bézout's identity | Coding theory | Euclidean algorithm | Finite field | Integer overflow | Certifying algorithm | Finite field arithmetic | The Art of Computer Programming | Greatest common divisor | Resultant | Additive inverse | Cryptography | Euclidean division | Monic polynomial | Unit (ring theory) | Field (mathematics) | Simple extension | Euclidean domain | Euclid's lemma | Prime number | Irreducible polynomial | Computer algebra | Polynomial greatest common divisor | Arithmetic | Modular arithmetic | Modular multiplicative inverse | Multiplicative inverse