In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers...

arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common...

The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, is an algorithm that computes the greatest common divisor...

are called integer division algorithms, the best known of which being long division. Euclidean division, and algorithms to compute it, are fundamental...

polynomials all the properties that may be deduced from the Euclidean algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties...

generalization of Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the...

a) = |a|. This case is important as the terminating step of the Euclidean algorithm. The above definition is unsuitable for defining gcd(0, 0), since...

in Voronoi diagrams. Although the algorithm may be applied most directly to the Euclidean plane, similar algorithms may also be applied to higher-dimensional...

RSA algorithm. A benefit for the computer implementation of these applications is that there exists a very fast algorithm (the extended Euclidean algorithm)...

Lehmer's GCD algorithm, named after Derrick Henry Lehmer, is a fast GCD algorithm, an improvement on the simpler but slower Euclidean algorithm. It is mainly...

numbers Euclidean domain, a ring in which Euclidean division may be defined, which allows Euclid's lemma to be true and the Euclidean algorithm and the...

popular algorithms for this task are: Peterson–Gorenstein–Zierler algorithm Berlekamp–Massey algorithm Sugiyama Euclidean algorithm Peterson's algorithm is...

The Euclidean rhythm in music was discovered by Godfried Toussaint in 2004 and is described in a 2005 paper "The Euclidean Algorithm Generates Traditional...

λ(n) = lcm(p − 1, q − 1). The lcm may be calculated through the Euclidean algorithm, since lcm(a, b) = ⁠|ab|/gcd(a, b)⁠. λ(n) is kept secret. Choose...

{p}}} . Once x {\displaystyle x} is determined, one can apply the Euclidean algorithm with p {\displaystyle p} and x {\displaystyle x} . Denote the first...

Gröbner bases. The Euclidean algorithm for computing the polynomial greatest common divisor is a special case of Buchberger's algorithm restricted to polynomials...

deterministic algorithm and within ( 33 + ε ) / 25 {\displaystyle (33+\varepsilon )/25} by a randomized algorithm. The TSP, in particular the Euclidean variant...

properties with integers: they form a Euclidean domain, and thus have a Euclidean division and a Euclidean algorithm; this implies unique factorization and...

is the minimum Euclidean distance assignment. Hartigan, J. A.; Wong, M. A. (1979). "Algorithm AS 136: A k-Means Clustering Algorithm". Journal of the...

decoding algorithm. In 1975, another improved BCH scheme decoder was developed by Yasuo Sugiyama, based on the extended Euclidean algorithm. In 1977,...

Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely...

numbers and other arithmetic-related concepts. Book 7 includes the Euclidean algorithm, a method for finding the greatest common divisor of two numbers...

unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such...

is the result of Gabriel Lamé's analysis of the complexity of the Euclidean algorithm. Using Fibonacci numbers, he proved in 1844 that when looking for...

modular multiplicative inverse d of b modulo m using the extended Euclidean algorithm. That is: c = be mod m = d−e mod m, where e < 0 and b ⋅ d ≡ 1 (mod...

fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number ⁠ p {\displaystyle...

extended Euclidean algorithm. To find the needed a {\displaystyle a} , b {\displaystyle b} , A {\displaystyle A} , and B {\displaystyle B} the algorithm uses...

coprime. It can be constructed using the extended Euclidean algorithm. The extended Euclidean algorithm efficiently determines integers R′ and N′ that satisfy...

Babylonia in 200 BC. Another ancient decrease-and-conquer algorithm is the Euclidean algorithm to compute the greatest common divisor of two numbers by...

Euclidean algorithm, as well as Farey sequences and a number of related mathematical constructs. Bresenham's line algorithm Circle drawing algorithm Rasterization...