Method for computing the relation of two integers with their greatest common divisor
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.
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arithmetic and computer programming, the extendedEuclideanalgorithm is an extension to the Euclideanalgorithm, and computes, in addition to the greatest...
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a quotient and a remainder Euclideanalgorithm, a method for finding greatest common divisors ExtendedEuclideanalgorithm, a method for solving the Diophantine...
but when an explicit algorithm for Euclidean division is known, one may use the Euclideanalgorithm and extendedEuclideanalgorithm to compute greatest...
RSA algorithm. A benefit for the computer implementation of these applications is that there exists a very fast algorithm (the extendedEuclidean algorithm)...
Sugiyama's adaptation of the ExtendedEuclideanalgorithm. Correction of unreadable characters could be incorporated to the algorithm easily as well. Let k 1...
identity. Numbers p and q like this can be computed with the extendedEuclideanalgorithm. gcd(a, 0) = |a|, for a ≠ 0, since any number is a divisor of...
m_{1}} and m 2 {\displaystyle m_{2}} may be computed by the extendedEuclideanalgorithm. A solution is given by x = a 1 m 2 n 2 + a 2 m 1 n 1 . {\displaystyle...
computed before the message is known. It may be computed using the extendedEuclideanalgorithm or using Fermat's little theorem as k q − 2 mod q {\displaystyle...
the modular multiplicative inverse d of b modulo m using the extendedEuclideanalgorithm. That is: c = be mod m = d−e mod m, where e < 0 and b ⋅ d ≡ 1...
polynomial GCD allows extending to univariate polynomials all the properties that may be deduced from the Euclideanalgorithm and Euclidean division. Moreover...
are coprime. It can be constructed using the extendedEuclideanalgorithm. The extendedEuclideanalgorithm efficiently determines integers R′ and N′ that...
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...
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are called integer division algorithms, the best known of which being long division. Euclidean division, and algorithms to compute it, are fundamental...
the modular multiplicative inverse can be computed using the extendedEuclideanalgorithm. An alternative is to compute s − 1 {\displaystyle s^{-1}} as...
deterministic algorithm and within ( 33 + ε ) / 25 {\displaystyle (33+\varepsilon )/25} by a randomized algorithm. The TSP, in particular the Euclidean variant...
{p}}\\m_{q}&=c^{{\frac {1}{4}}(q+1)}{\bmod {q}}\end{aligned}}} Use the extendedEuclideanalgorithm to find y p {\displaystyle y_{p}} and y q {\displaystyle y_{q}}...
The Euclidean rhythm in music was discovered by Godfried Toussaint in 2004 and is described in a 2005 paper "The EuclideanAlgorithm Generates Traditional...
multiplication by the inverse modulo p, which may be computed using the extendedEuclideanalgorithm. A particular case is GF(2), where addition is exclusive OR (XOR)...
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