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Topic: Extended Euclidean algorithm

	PlanetMath: fast Euclidean algorithm
		The algorithm can also be used to compute any particular pair of coefficients from the Extended Euclidean Algorithm, although computing every pair of coefficients would involve
		The full algorithm, and a comprehensive runtime analysis is given in “Modern Computer Algebra” by von zur Gathen and Gerhard.
		This is version 4 of fast Euclidean algorithm, born on 2004-07-22, modified 2005-04-14.
	www.planetmath.org /encyclopedia/FastEuclideanAlgorithm.html (222 words)

	PlanetMath: Berlekamp-Massey algorithm
		The Berlekamp-Massey algorithm is used for finding the minimal polynomial of a linearly recurrent sequence.
		The algorithm itself is presented at the end of this article.
		This is version 4 of Berlekamp-Massey algorithm, born on 2004-07-22, modified 2005-04-14.
	planetmath.org /encyclopedia/BerlekampMasseyAlgorithm.html (0 words)

	Random Works of the Web » Blog Archive » Extended Euclidean algorithm (Site not responding. Last check: )
		The extended Euclidean algorithm is an algorithm used to calculate the greatest common divisor (gcd, or also highest common factor, HCF) of two integers a and b, as well as integers x and y such that
		To illustrate the extension of the Euclid’s algorithm, consider the computation of gcd(120, 23), which is shown on the table on the left.
		This method attempts to solve the required equation with the sum replaced by the remainders in each step of the algorithm, which is larger than their gcd, but are decreasing in magnitude, and so will eventually become the required equation.
	random.dragonslife.org /extended-euclidean-algorithm/4414 (0 words)

	Euclidean algorithm - Wikipedia, the free encyclopedia
		In number theory, the Euclidean algorithm (also called Euclid's algorithm) is an algorithm to determine the greatest common divisor (GCD) of two integers or elements of any Euclidean domain (for example, polynomials over a field) by repeatedly dividing the two numbers and the remainder in turns.
		This is known as the extended Euclidean algorithm.
		The quotients that appear when the Euclidean algorithm is applied to the inputs a and b are precisely the numbers occurring in the continued fraction representation of a/b.
	en.wikipedia.org /wiki/Euclidean_algorithm (0 words)

	Extended Euclidean algorithm - Wikipedia, the free encyclopedia
		The extended Euclidean algorithm is an extension to the Euclidean algorithm for finding the greatest common divisor (GCD) of a and b: it also finds the integers x and y in Bezout's identity
		The extended Euclidean algorithm is particularly useful when a and b are coprime, since x is the multiplicative inverse of a modulo b.
		The extended Euclidean algorithm can also be used to calculate the multiplicative inverse in a finite field.
	en.wikipedia.org /wiki/Extended_Euclidean_algorithm (0 words)

	Extended Euclidean Algorithm
		You repeatedly divide the divisor by the remainder until the remainder is 0.
		This is known as the extended Euclidean Algorithm.
		Before presenting this extended Euclidean algorithm, we shall look at a special application that is the most common usage of the algorithm.
	www-math.cudenver.edu /~wcherowi/courses/m5410/exeucalg.html (718 words)

	Euclidean algorithm used to decode Reed-Solomon codes? \| Comp.DSP \| DSPRelated.com
		Euclidean algorithm is based, or perhaps give some pointers in
		Yes, it is the same algorithm or more precisely, the extended
		Euclidean algorithm for finding the GCD of two polynomials
	www.dsprelated.com /showmessage/22571/1.php (636 words)

	Math 413 Lecture 2 - Divisibility & Euclidean Algorithm
		This is useful both because the Euclidean algorithm is the primary computational tool in number theory and because the existence of a Euclidean algorithm has strong consequences for the structure of the ring.
		The existence of a Euclidean algorithm is sufficiently important that we call a ring with such an algorithm a Euclidean ring.
		Careful analyses of the Euclidean algorithm and faster variants of the algorithm can be found in [Knuth2] and [BS96].
	www.math.umbc.edu /~campbell/Math413Spr01/Lectures/lecture2.html (455 words)

	PlanetMath: fast Euclidean algorithm
		The algorithm can also be used to compute any particular pair of coefficients from the Extended Euclidean Algorithm, although computing every pair of coefficients would involve
		The full algorithm, and a comprehensive runtime analysis is given in “Modern Computer Algebra” by von zur Gathen and Gerhard.
		This is version 4 of fast Euclidean algorithm, born on 2004-07-22, modified 2005-04-14.
	planetmath.org /encyclopedia/FastEuclideanAlgorithm.html (0 words)

	[No title] (Site not responding. Last check: )
		Use the extended Euclidean algorithm to solve g = mx + ny for a pair of integers m and n.
		Describe an algorithm to convert a base 10 fraction of the form m / n where m and n are small positive integers to base b.
		Describe an algorithm to convert a fraction with denominator that is a power of two to base two.
	www.cbu.edu /~yanushka/j0/r.0 (0 words)

	Euclidean Algorithm -- from Wolfram MathWorld
		There are even principal rings which are not Euclidean but where the equivalent of the Euclidean algorithm can be defined.
		The Euclidean algorithm is an example of a P-problem whose time complexity is bounded by a quadratic function of the length of the input values (Bach and Shallit 1996).
		remainders, so the algorithm can be easily applied by hand by repeatedly computing remainders of consecutive terms starting with the two numbers of interest (with the larger of the two written first).
	mathworld.wolfram.com /EuclideanAlgorithm.html (0 words)

	re: brain teaser #64
		I imagine that Nate's approach is closely related to the approach using the extended euclidean algorithm.
		The trick to the extended euclidean algorithm is to remember the steps you performed while computing the gcd.
		The first step of the euclidean algorithm for computing this gcd is to subtract from 109 a multiple of 37, preferably the largest multiple of 37 that is less than 109, which is 74.
	www.physicsforums.com /showthread.php?p=96495 (0 words)

	S2.html
		First of all the algorithm terminates since by the division equation thereom, the remainders form a decreasing sequence of non-negative integers.
		Since these equations form a k-step Eucldean algorithm for inputs r[0] and r[1], use the induction assumption to assume that they can be back-solved for the integers x and y satiifying the equation xr[0]+yr[1]=r[k]=g=gcd(r[0],r[1]).
		We use the extended Euclidean algorithm on 48 and 37.
	www.math.sfu.ca /~gfee/Math342/A2/S21.html (0 words)

	Euclidean algorithm and its applications
		The Euclidean algorithm is simply a sequence of long divisions (repeated until the remainder becomes zero).
		The following extended Euclidean algorithm (EEA) also expresses each remainder as a linear combination of the original polynomials.
		EEA is fine to use for finite fields, since the coefficients of the intermediate polynomials never grow big.
	www.math.clemson.edu /faculty/Gao/calg/node5.html (306 words)

	Cryptography Tutorial - The Euclidean Algorithm finds the Greatest Common Divisor of two Integers
		1) Understand the Extended Euclidean Algorithm to determine the inverse of a given integer.
		Consequently, if a and b have a greatest common divisor different from 1 (that is the gcd(a,b) is not 1) a does not have an inverse mod b.
		Again, the Extended Euclidean Algorithm should be performed by a computer as it is very easy to implement and it yields the answer quickly.
	www.antilles.k12.vi.us /math/cryptotut/extended_euclidean_algorithm.htm (0 words)

	Extended Euclidean algorithm
		The extended Euclidean algorithm is a version of the Euclidean algorithm; its input are two integers a and b and the algorithm computes their greatest common divisor (GCD) as well as integers x and y such that ax + by = gcd(a,b).
		The extended Euclidian algorithm can also be used to calculate the multiplicative inverse in a finite field.
		Given the irreducible polynomial f(x) used to define the finite field, and the element a(x) whose inverse is desired, then a form of the algorithm suitable for determining the inverse is given by the following pseudocode:
	www.fact-index.com /e/ex/extended_euclidean_algorithm.html (0 words)

	[No title] (Site not responding. Last check: )
		CS 236 Extended Euclidean & RSA Algorithms 20 Feb. For positive integers M and N, find the greatest common divisor G of M and N and find integers x and y of opposite signs with Mx + Ny = G efficiently.
		RSA Algorithm RSA denotes the first letters of the last names of the inventors of the algorithm, namely, Ronald L. Rivest, Adi Shamir and Leonard Adleman.
		The RSA algorithm depends on difficulty of factoring a large integer into two primes and on the ease of finding large primes.
	www.cbu.edu /~yanushka/od/n.0 (0 words)

	Euclidean algorithm
		The Euclidean algorithm is a way to find the greatest common divisor of two positive integers, a and b.
		The greatest common divisor of 210 and 45 is 15, and we have written 15 as a sum of integer multiples of 210 and 45.
		The extended Euclidean algorithm is easy to implement on a computer and the amount of memory needed is not large.
	www.math.rutgers.edu /~greenfie/gs2004/euclid.html (793 words)

	Polynomials - HowTo: Euclidean algorithm for polynomials
		In this HowTo we will describe the analogue of the euclidean algorithm to compute the greatest common divisor of any two polynomials in
		The euclidean algorithm then works in the same way.
		One such condition is that the gcd be monic; that is, that the coefficient of highest degree be
	xmlearning.maths.ed.ac.uk /lecture_notes/polynomials/howto_euclidean_algorithm_polynomials/howto_euclidean_algorithm_polynomials.php (202 words)

	Hidden Small Exponent Method Information (Site not responding. Last check: )
		However, it is a simple matter to decrypt messages that have been encrypted using pseudo keys generated by the hidden exponent algorithm.
		We invite you to learn about this algorithm, try out the 128-bit key generator (details on implementation provided) and try the official challenge yourself.
		Now that the RSA algorithm is public domain, companies no longer need to license it from RSA, leaving the purity of the implementation open to "interpretations".
	crypto.cs.mcgill.ca /~crepeau/RSA (294 words)

	A Matrix Interpretation of the Extended Euclidean Algorithm (ResearchIndex) (Site not responding. Last check: )
		The extended Euclidean algorithm for polynomials and formal power series that is used for the recursive computation of Pade approximants can be viewed in various ways as a sequence of successive matrix multiplications that are applied to a Sylvester matrix with the original data.
		Here we present this result in a general version that includes the treatment of the Cabay{Meleshko look-ahead algorithm, which generalizes the extended Euclidean algorithm and yields a weakly stable (forward stable)...
		1 A weakly stable algorithm for Pade approximants and the inv..
	citeseer.ist.psu.edu /310864.html (0 words)

	[No title]
		CS 122 Extended Euclidean algorithm 30 Mar. I feel some of you did not take notes yesterday.
		Find the greatest command divisor of the positive integers a and b as g and then solve the equation g = x * a + y * b for integers x and y of opposite signs.
		Modify Euclid's algorithm to find the greatest common divisor to compute and save the quotients as an array.
	www.cbu.edu /~yanushka/j1/n.0 (0 words)

	The Laws of Cryptography: Cryptographers' Favorite Algorithms
		It's possible to code the extended gcd algorithm following the model above, first using a loop to calculate the gcd, while saving the quotients at each stage, and then using a second loop as above to work back through the equations, solving for the gcd in terms of the original two numbers.
		An essential part of many of the algorithms involved is to raise an integer to another integer power, modulo an integer (taking the remainder on division).
		It is very similar to the previous algorithm, but differs in processing the binary bits of the exponent in the opposite order.
	www.cs.utsa.edu /~wagner/laws/fav_alg.html (0 words)

	[No title] (Site not responding. Last check: )
		The Extended Euclidean Algorithm -------------------------------- There are two prevailing versions of expositions of the extended Euclidean algorithm.
		This is where we extend the Euclidean algorithm.
		Because this process is just the plain old Euclidean algorithm when you just look at the values of w,z, eventually one of them, say w, will become 0, and z will become gcd(a,b): au + bv = 0 ax + by = gcd(a,b) Then you take the second equation and you are done!
	www.cs.toronto.edu /~trebla/ExtendedEuclid.txt (0 words)

	PHP Bugs: #15835: Problem on function gmp_invert ()
		I implemented the extended Euclides algoritm that do exactly the same that this function and, it returned me the right value.
		The Euclidean Algorithm computes the greatest common divisor, gmp_invert doesn't seem to do that...
		The "Euclidean Algorithm" computes the greatest common divisor but, the "Extended Euclidean Algorithm" give you the multiplicative inverse.
	bugs.php.net /15835 (0 words)

	Euclidean algorithm algorithm wszystko o google- cybersocjologia (Site not responding. Last check: )
		The Euclidean algorithm (also called Euclid's algorithm) is an algorithm to determine the greatest common divisor (gcd) of two integers.
		The proof of this algorithm is not difficult.
		When analyzing the runtime of Euclid's algorithm, it turns out that the inputs requiring the most divisions are two successive Fibonacci numbers, and the worst case requires ¥È(n) divisions, where n is the number of digits in the input (see Big O notation).
	www.ifb.pl /~pneuma/old/archiwum-google.php?lng=en&pg=322 (0 words)

	On the Complexity of the Extended Euclidean Algorithm (Extended Abstract) (ResearchIndex) (Site not responding. Last check: )
		On the Complexity of the Extended Euclidean Algorithm (Extended Abstract) (ResearchIndex)
		On the Complexity of the Extended Euclidean Algorithm (Extended Abstract)
		2 A new algorithm and re ned bounds for extended GCD computati..
	citeseer.ist.psu.edu /641687.html (0 words)

	Footnotes (Site not responding. Last check: )
		The naming convention is lower case sans-serif for example generators of cryptosystems and initial capitals for actual cryptosystems e.g.
		This refers to using the wrong result from performing the extended euclidean algorithm on K
		This refers to using the wrong result from performing the extended euclidean algorithm on weight w
	www.csse.monash.edu.au /~skcho5/Thesis/footnode.html (0 words)

	An Example Using the Extended Euclidean Algorithm (Site not responding. Last check: )
		If you're using the algorithm by hand (as opposed to using a computer), you can use a table to keep the numbers straight and remember what to do.
		I'll illustrate by applying the algorithm to find the greatest common divisor of 187 and 102.
		The statement and proof of the Extended Euclidean Algorithm
	marauder.millersville.edu /~bikenaga/absalg/exteuc/exteucex.html (209 words)

	The Extended Euclidean Algorithm
		for the Euclidean algorithm for m/n are printed.
		The length l of the algorithm is printed.
		The continued fraction for -m/n is also printed.
	www.numbertheory.org /php/euclid.html (96 words)

	Extended Euclidean Algorithm
		Enter 2 positive integers to find their G.C.D using the extended Euclidean Algorithm
		Please note : This script has been written to show the Extended Euclidean Algorithm.
		No guarantee can be given over the validity of the answers.
	homepages.inf.ed.ac.uk /s0563270/maths/php/euclidean.php (73 words)

	The Euclidean algorithm
		The division algorithm (theorem 1.2.7) tells us that there are integers
		The proof of this is constructive and follows an algorithm called the Euclidean algorithm.
		Use the division algorithm (theorem 1.2.7) to compute integers
	web.usna.navy.mil /~wdj/book/node13.html (217 words)

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