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Method for computing the relation of two integers with their greatest common divisor
arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest
Extended_Euclidean_algorithm
Algorithm for computing greatest common divisors
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers
Euclidean_algorithm
Concept in modular arithmetic
RSA algorithm. A benefit for the computer implementation of these applications is that there exists a very fast algorithm (the extended Euclidean algorithm)
Modular multiplicative inverse
Modular_multiplicative_inverse
Commutative ring with a Euclidean division
ring of integers), but lacks an analogue of the Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors. So, given an
Euclidean_domain
Algorithm for computing the greatest common divisor
The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, is an algorithm that computes the greatest common divisor
Binary_GCD_algorithm
Algorithm for fast modular multiplication
are coprime. It can be constructed using the extended Euclidean algorithm. The extended Euclidean algorithm efficiently determines integers R′ and N′ that
Montgomery modular multiplication
Montgomery_modular_multiplication
Error correction code
Sugiyama's adaptation of the Extended Euclidean algorithm. Correction of unreadable characters could be incorporated to the algorithm easily as well. Let k 1
BCH_code
Topics referred to by the same term
a quotient and a remainder Euclidean algorithm, a method for finding greatest common divisors Extended Euclidean algorithm, a method for solving the Diophantine
Euclidean
Digital verification standard
computed before the message is known. It may be computed using the extended Euclidean algorithm or using Fermat's little theorem as k q − 2 mod q {\displaystyle
Digital_Signature_Algorithm
Relating two numbers and their greatest common divisor
called Bézout coefficients for (a, b); they are not unique. The extended Euclidean algorithm can be used to compute a minimal pair of Bézout coefficients
Bézout's_identity
Largest integer that divides given integers
identity. Numbers p and q like this can be computed with the extended Euclidean algorithm. gcd(a, 0) = |a|, for a ≠ 0, since any number is a divisor of
Greatest_common_divisor
Mathematical algorithm
{n}}} . Solutions to this equation are easily obtained using the extended Euclidean algorithm. To find the needed a {\displaystyle a} , b {\displaystyle b}
Pollard's rho algorithm for logarithms
Pollard's_rho_algorithm_for_logarithms
About simultaneous modular congruences
m_{1}} and m 2 {\displaystyle m_{2}} may be computed by the extended Euclidean algorithm. A solution is given by x = a 1 m 2 n 2 + a 2 m 1 n 1 . {\displaystyle
Chinese_remainder_theorem
Algorithm for public-key cryptography
de ≡ 1 (mod λ(n)); d can be computed efficiently by using the extended Euclidean algorithm, since, thanks to e and λ(n) being coprime, said equation is
RSA_cryptosystem
Fast greatest common divisor algorithm
GCD algorithm, named after D. H. Lehmer, is a fast GCD algorithm for multiple-precision arithmetic, which improves on the simpler Euclidean algorithm by
Lehmer's_GCD_algorithm
Algebraic structure
may be computed by using the extended Euclidean algorithm (see Modular multiplicative inverse § Extended Euclidean algorithm). Let F {\displaystyle F} be
Finite_field
Error-correcting codes
decoding algorithm. In 1975, another improved BCH scheme decoder was developed by Yasuo Sugiyama, based on the extended Euclidean algorithm. In 1977,
Reed–Solomon_error_correction
Mathematical algorithm
Kuṭṭaka algorithm has much similarity with and can be considered as a precursor of the modern day extended Euclidean algorithm. The latter algorithm is a
Kuṭṭaka
NP-hard problem in combinatorial optimization
deterministic algorithm and within ( 33 + ε ) / 25 {\displaystyle (33+\varepsilon )/25} by a randomized algorithm. The TSP, in particular the Euclidean variant
Travelling_salesman_problem
Dijkstra notation with non-deterministic conditionals
b hold the greatest common divisor of A and B. Dijkstra sees in this algorithm a way of synchronizing two infinite cycles a := a - b and b := b - a in
Guarded_Command_Language
Exponentation in modular arithmetic
multiplicative inverse d of b modulo m (for instance by using extended Euclidean algorithm). More precisely: c = be mod m = d−e mod m, where e < 0 and b
Modular_exponentiation
Division with remainder of integers
are called integer division algorithms, the best known of which being long division. Euclidean division, and algorithms to compute it, are fundamental
Euclidean_division
Maximally even rhythm
The Euclidean rhythm in music was discovered by Godfried Toussaint in 2004 and is described in a 2005 paper "The Euclidean Algorithm Generates Traditional
Euclidean_rhythm
Algorithm used for points in euclidean space
in Voronoi diagrams. Although the algorithm may be applied most directly to the Euclidean plane, similar algorithms may also be applied to higher-dimensional
Lloyd's_algorithm
calculus algorithm Pohlig–Hellman algorithm Pollard's rho algorithm for logarithms Euclidean algorithm: computes the greatest common divisor Extended Euclidean
List_of_algorithms
'Best' approximation of a function by a rational function of given order
divergent series. One way to compute a Padé approximant is via the extended Euclidean algorithm for the polynomial greatest common divisor. The relation R (
Padé_approximant
Overview of and topical guide to algorithms
Regular expression Parsing Earley parser CYK algorithm Euclidean algorithm Extended Euclidean algorithm Sieve of Eratosthenes Integer factorization Primality
Outline_of_algorithms
Vector quantization algorithm minimizing the sum of squared deviations
is the minimum Euclidean distance assignment. Hartigan, J. A.; Wong, M. A. (1979). "Algorithm AS 136: A k-Means Clustering Algorithm". Journal of the
K-means_clustering
Problem of inverting exponentiation in groups
means congruence modulo p {\displaystyle p} in the integers. The extended Euclidean algorithm finds k {\displaystyle k} quickly. With Diffie–Hellman, a cyclic
Discrete_logarithm
A prime p divides a^p–a for any integer a
values of y, e and n is easy if one knows φ(n). In fact, the extended Euclidean algorithm allows computing the modular inverse of e modulo φ(n), that is
Fermat's_little_theorem
planar by a certifying algorithm that outputs either a planar embedding or a Kuratowski subgraph. The extended Euclidean algorithm for the greatest common
Certifying_algorithm
Greatest common divisor of polynomials
polynomial GCD allows extending to univariate polynomials all the properties that may be deduced from the Euclidean algorithm and Euclidean division. Moreover
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Representation of modular integers by "small" fractions
complexity of the Euclidean algorithm. More precisely, given the two integers m and a appearing in Thue's lemma, the extended Euclidean algorithm computes three
Thue's_lemma
Public-key cryptosystem
the modular multiplicative inverse can be computed using the extended Euclidean algorithm. An alternative is to compute s − 1 {\displaystyle s^{-1}} as
ElGamal_encryption
topics named after the Greek mathematician Euclid. Euclidean algorithm Extended Euclidean algorithm Euclidean division Euclid–Euler theorem Euclid number Euclid's
List of things named after Euclid
List_of_things_named_after_Euclid
Form of public key cryptography
of r {\displaystyle r} modulo q {\displaystyle q} using the Extended Euclidean algorithm. The inverse will exist since r {\displaystyle r} is coprime
Merkle–Hellman knapsack cryptosystem
Merkle–Hellman_knapsack_cryptosystem
Number which when multiplied by x equals 1
inverse of 3 mod 11 is four because 4 ⋅ 3 ≡ 1 (mod 11). The extended Euclidean algorithm may be used to compute it. The sedenions are an algebra in which
Multiplicative_inverse
Computation modulo a fixed integer
solving Bézout's equation a x + m y = 1 for x, y, by using the Extended Euclidean algorithm. In particular, if p is a prime number, then a is coprime with
Modular_arithmetic
Arithmetic in a field with a finite number of elements
multiplication by the inverse modulo p, which may be computed using the extended Euclidean algorithm. A particular case is GF(2), where addition is exclusive OR (XOR)
Finite_field_arithmetic
Algorithm for finding shortest paths
path problem. A* search algorithm Bellman–Ford algorithm Euclidean shortest path Floyd–Warshall algorithm Johnson's algorithm Longest path problem Parallel
Dijkstra's_algorithm
Kind of error correction code
{\displaystyle a(x)} and b ( x ) {\displaystyle b(x)} using the extended euclidean algorithm, so that a ( x ) ≡ b ( x ) ⋅ v ( x ) mod g ( x ) {\displaystyle
Binary_Goppa_code
One over a whole number
fraction can be converted into an equivalent whole number using the extended Euclidean algorithm. This conversion can be used to perform modular division: dividing
Unit_fraction
Method for division with remainder
result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software. Division algorithms fall into
Division_algorithm
Cryptographic algorithm created by Adi Shamir
A is B such that A*B % p == 1). This can be computed via the extended Euclidean algorithm http://en.wikipedia.org/wiki/Modular_multiplicative_inverse#Computation
Shamir's_secret_sharing
English mathematician (1682–1716)
he had lived we would have known something." Cotes's spiral Extended Euclidean algorithm Newton–Cotes formulas Lituus (mathematics) Gowing 2002, p. 5
Roger_Cotes
Topics referred to by the same term
the twin study Ethylene-ethyl acid, used in hot-melt adhesive Extended Euclidean algorithm Extreme event attribution, the science of quantifying the degree
EEA_(disambiguation)
Type of plane partition
of points { p 1 , … p n } {\displaystyle \{p_{1},\dots p_{n}\}} in the Euclidean plane. In this case, each point p k {\displaystyle p_{k}} has a corresponding
Voronoi_diagram
Shortest network connecting points
A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-dimensional Euclidean space connects the points by a system
Euclidean minimum spanning tree
Euclidean_minimum_spanning_tree
Least common multiple Euclidean algorithm Coprime Euclid's lemma Bézout's identity, Bézout's lemma Extended Euclidean algorithm Table of divisors Prime
List_of_number_theory_topics
Shape with three sides
generally, four points in three-dimensional Euclidean space determine a solid figure called tetrahedron. In non-Euclidean geometries, three "straight" segments
Triangle
Algebraic structure
divisor that is monic (leading coefficient equal to 1). The extended Euclidean algorithm allows computing (and proving) Bézout's identity. In the case
Polynomial_ring
Mathematical puzzle
Coconuts, a copy of the story as it appeared in the Saturday Evening Post The Monkey and the Coconuts: An Introduction to the Extended Euclidean Algorithm
The_monkey_and_the_coconuts
develops Kuṭṭaka, an algorithm very similar to the Extended Euclidean algorithm. 499: Aryabhata describes a numerical algorithm for finding cube roots
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
Algorithm for integer factorization
residue classes modulo n {\displaystyle n} , performed using the extended Euclidean algorithm. In particular, division by some v mod n {\displaystyle v{\bmod
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
Finding the smallest circle that contains all given points
Welzl's minidisk algorithm has been extended to handle Bregman divergences which include the squared Euclidean distance. Megiddo's algorithm is based on the
Smallest-circle_problem
On short connecting nets with added points
the Euclidean Steiner tree problem is NP-hard, and hence it is not known whether an optimal solution can be found by using a polynomial-time algorithm. However
Steiner_tree_problem
Type of algebraic curve
over the field K {\displaystyle K} . The algorithm works as follows Using the extended Euclidean algorithm compute the polynomials d 1 , e 1 , e 2 ∈
Imaginary_hyperelliptic_curve
Public-key encryption scheme
{p}}\\m_{q}&=c^{{\frac {1}{4}}(q+1)}{\bmod {q}}\end{aligned}}} Use the extended Euclidean algorithm to find y p {\displaystyle y_{p}} and y q {\displaystyle y_{q}}
Rabin_cryptosystem
Some polynomial Diophantine equations can be solved using the extended Euclidean algorithm, which works as well with polynomials as it does with integers
Polynomial Diophantine equation
Polynomial_Diophantine_equation
{\displaystyle a} and b {\displaystyle b} will be integers. Using the Extended Euclidean Algorithm, compute the inverse of b {\displaystyle b} modulo p {\displaystyle
Okamoto–Uchiyama_cryptosystem
Result on periodic sequences
theorem above. The proof comes from, and is closely related to the extended Euclidean algorithm, much like the proof of Bézout's identity. Let u , v {\displaystyle
Fine_and_Wilf's_theorem
Statistical method in data analysis
cluster. At each step, the algorithm merges the two most similar clusters based on a chosen distance metric (e.g., Euclidean distance) and linkage criterion
Hierarchical_clustering
Mathematical model of the physical space
Euclidean geometry is a mathematical system attributed to Euclid, an ancient Greek mathematician, which he described in his textbook on geometry, Elements
Euclidean_geometry
Density-based data clustering algorithm
HDBSCAN* algorithm. pyclustering library includes a Python and C++ implementation of DBSCAN for Euclidean distance only as well as OPTICS algorithm. SPMF
DBSCAN
ring of univariate polynomials over a field. In this case, the extended Euclidean algorithm may be used for computing the above unimodular matrix; see Polynomial
Linear_equation_over_a_ring
Triangulation method
of Delaunay triangulation extends to three and higher dimensions. Generalizations are possible to metrics other than Euclidean distance. However, in these
Delaunay_triangulation
Approach to quantum gravity utilizing Wick rotations
In theoretical physics, Euclidean quantum gravity exploits the Wick rotation to describe gravity according to the principles of quantum mechanics. This
Euclidean_quantum_gravity
British mathematician and cryptographer
Encryption Using a Finite Field" (A couple of typos in this pdf: Extended Euclidean Algorithm modulus should be (p-1) instead of p. Enc and Dec are performed
Malcolm_J._Williamson
Indian inventions
Kuṭṭaka algorithm has much similarity with and can be considered as a precursor of the modern day extended Euclidean algorithm. The latter algorithm is a
List of Indian inventions and discoveries
List_of_Indian_inventions_and_discoveries
exist positive integers ri and si, that can be found using the Extended Euclidean algorithm, such that r i . m i + s i . M / m i = 1 {\displaystyle r_{i}
Secret sharing using the Chinese remainder theorem
Secret_sharing_using_the_Chinese_remainder_theorem
Area of discrete mathematics
graph is the special case of a Euclidean graph. The Euclidean graph allows its edges to have the length of the Euclidean distance between its endpoints
Graph_theory
Partition of the Euclidean plane
tesselation or a sectional Dirichlet tesselation, is a partition of the Euclidean plane into polygonal cells defined from a set of circles. The cell for
Power_diagram
Root-finding algorithm for polynomials
p-f_{0}g_{0}=f_{0}\Delta g+g_{0}\Delta f} using any variant of the extended Euclidean algorithm to obtain the incremented approximations f 1 = f 0 + Δ f {\displaystyle
Splitting_circle_method
Arithmetic operation
integers as result, is sometimes called Euclidean division, because it is the basis of the Euclidean algorithm. Give the integer quotient as the answer
Division_(mathematics)
Decomposition of a number into a product
efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty
Integer_factorization
Mathematical treatise by Euclid
Euclidean geometry, elementary number theory, and incommensurability. These include the Pythagorean theorem, Thales' theorem, the Euclidean algorithm
Euclid's_Elements
Algorithm for supervised learning of binary classifiers
In machine learning, the perceptron is an algorithm for supervised learning of binary classifiers. A binary classifier is a function that can decide whether
Perceptron
Proof that a number is prime
calculation of gcd, done for large numbers usually using the Extended Euclidean algorithm, over the number of primes provided. Each operation takes between
Primality_certificate
to that of the Euclidean minimum spanning tree. Although known construction methods for them are slow, fast approximation algorithms with similar properties
Greedy_geometric_spanner
Branch of mathematics
geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance,
Geometry
Formula for the "volume" of an n-simplex
theorem comes from the following algorithm for realizing a Euclidean Distance Matrix or a Gramian Matrix. Input Euclidean Distance Matrix Δ {\displaystyle
Cayley–Menger_determinant
Condition under which an odd prime is a sum of two squares
{p}}} . Once x {\displaystyle x} is determined, one can apply the Euclidean algorithm with p {\displaystyle p} and x {\displaystyle x} . Denote the first
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Physical simulation to visualize graphs
Force-directed graph drawing algorithms are a class of algorithms for drawing graphs in an aesthetically-pleasing way. Their purpose is to position the
Force-directed_graph_drawing
Mathematical space with a notion of distance
real line. Arthur Cayley, in his article "On Distance", extended metric concepts beyond Euclidean geometry into domains bounded by a conic in a projective
Metric_space
Axiom set used in first-order logic
b\dots .} This fact allowed Tarski to prove that Euclidean geometry is decidable: there exists an algorithm which can determine the truth or falsity of any
Tarski's_axioms
Graph of intervisible locations in computational geometry
graphs have therefore been extended to the realm of time series analysis. Visibility graphs may be used to find Euclidean shortest paths among a set of
Visibility_graph
Feature selection algorithm used in binary classification
Relief is an algorithm developed by Kenji Kira and Larry Rendell in 1992 that takes a filter-method approach to feature selection that is notably sensitive
Relief_(feature_selection)
Property of a mathematical space
required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a
Dimension
Smallest convex set containing a given set
of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its
Convex_hull
Topological space that locally resembles Euclidean space
mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional
Manifold
Objects maximally similar to other objects in a dataset
k-medoids clustering algorithm, which is similar to the k-means algorithm but works when a mean or centroid is not definable. This algorithm basically works
Medoid
Asymmetric key encryption algorithm
mod q {\displaystyle u_{q}=x^{d_{q}}{\bmod {q}}} . Using the Extended Euclidean Algorithm, compute r p {\displaystyle r_{p}} and r q {\displaystyle r_{q}}
Blum–Goldwasser_cryptosystem
Optimization problem in computer science
version of the SVP under the Euclidean norm, several different approaches are known, which can be split into two classes: algorithms requiring superexponential
Lattice_problem
localization and mapping (SLAM) and relative position tracking, the algorithm was extended to 3D point clouds and has wide applications in computer vision
Normal distributions transform
Normal_distributions_transform
Method for mathematical optimization
optimization, the criss-cross algorithm is any of a family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general
Criss-cross_algorithm
Feature detection algorithm in computer vision
required for finding the Euclidean-distance-based nearest neighbor, an approximate algorithm called the best-bin-first algorithm is used. This is a fast
Scale-invariant feature transform
Scale-invariant_feature_transform
Computational geometry and optimization concept
In computational geometry and approximation algorithms, a coreset is a small, possibly weighted subset of an input point set that approximately preserves
Coreset
Smooth manifold with an inner product on each tangent space
such as distance, angles, length, volume, and curvature are defined. Euclidean space, the n {\displaystyle n} -sphere, hyperbolic space, and smooth surfaces
Riemannian_manifold
Nonlinear dimensionality reduction method
Dijkstra's algorithm, for example). The top n eigenvectors of the geodesic distance matrix, represent the coordinates in the new n-dimensional Euclidean space
Isomap
Optimization algorithm
unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function. The idea is to
Gradient_descent
EXTENDED EUCLIDEAN-ALGORITHM
EXTENDED EUCLIDEAN-ALGORITHM
Boy/Male
Muslim/Islamic
Servant of the Extender and Creator
Boy/Male
Hindu
Constisting of extended troops
Girl/Female
Indian, Sanskrit
Awakened; Roused; Expanded
Boy/Male
Indian
Servant of the expander, Extender
Biblical
burning; adoration,extended land
Boy/Male
Muslim
Intended, Aimed at, Object, Proposed
Girl/Female
Australian, Biblical, British, Christian, English, German, Hawaiian, Hebrew
Large; Extended; Broad; Spacious; Wide
Boy/Male
Arabic, German, Muslim
Intended; Proposed
Surname or Lastname
English
English : extended form of Yates.
Girl/Female
Muslim
Intended, Destined
Boy/Male
Tamil
Constisting of extended troops
Girl/Female
Arabic, Muslim
Intended; Destined
Boy/Male
Afghan, Arabic, Pashtun
Intended; Proposed
Boy/Male
Muslim
Servant of the expander, Extender
Boy/Male
Muslim
Servant of the Extender, Creator.
Boy/Male
Hindu, Indian, Marathi
Continuous Extended
Girl/Female
Biblical
Large; extended (name of a woman).
Boy/Male
Muslim
Servant of the Extender, Creator.
Boy/Male
Arabic
Servant of the Extender; Creator
Biblical
large; extended (name of a woman)
EXTENDED EUCLIDEAN-ALGORITHM
EXTENDED EUCLIDEAN-ALGORITHM
Girl/Female
Arabic, Australian, Muslim
Wife of Hazrat Ibrahim
Girl/Female
Latin
Protectress of sick children.
Girl/Female
Scandinavian
Ever kingly. Feminine of Eric.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Vaikuntam; The Abode of Lord Vishnu
Girl/Female
Indian
Victory
Surname or Lastname
English
English : variant spelling of Wilkinson.
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu
Eyes
Male
English
Anglicized form of Hebrew Yehowshaphat, JEHOSHAPHAT means "God has judged" or "whom God judges." In the bible, this is the name of many characters, including a king of Judah.
Girl/Female
Hindu
Name of a star, Well starred, From the Nakshatra Kritika
Boy/Male
Biblical American Hebrew
Poor, humble.
EXTENDED EUCLIDEAN-ALGORITHM
EXTENDED EUCLIDEAN-ALGORITHM
EXTENDED EUCLIDEAN-ALGORITHM
EXTENDED EUCLIDEAN-ALGORITHM
EXTENDED EUCLIDEAN-ALGORITHM
v. t.
To enlarge, as a surface or volume; to expand; to spread; to amplify; as, to extend metal plates by hammering or rolling them.
v. t.
To bestow; to offer; to impart; to apply; as, to extend sympathy to the suffering.
a.
Betrothed; affianced; as, an intended husband.
a.
Purposed; designed; as, intended harm or help.
n.
Extended area.
a.
Extended.
v. t.
To stretch out; to prolong in space; to carry forward or continue in length; as, to extend a line in surveying; to extend a cord across the street.
a.
Extended in length; tiresome.
a.
Extended horizontally; stretched out.
a.
Capable of being extended, susceptible of being stretched, extended, enlarged, widened, or expanded.
a.
Not extended.
v. t.
To increase in quantity by weakening or adulterating additions; as, to extend liquors.
n.
Related to Euclid, or to the geometry of Euclid.
adv.
In an extended manner.
n.
One who, or that which, extends or stretches anything.
v. t.
Outreaching; expansive; extended, superficially or otherwise.
v. t.
To enlarge; to widen; to carry out further; as, to extend the capacities, the sphere of usefulness, or commerce; to extend power or influence; to continue, as time; to lengthen; to prolong; as, to extend the time of payment or a season of trail.
a.
Made tense; stretched out; extended; forcible; violent.
a.
Drawn out; extended.
imp. & p. p.
of Extend