The teaching of geometry has been in crisis in america for over thirty years. There are two main differences: Thus, for saving memory, each indexed variable must be replaced by only two variables. Indeed, if a a 0d and b b0d for some integers a0 and b, then a. Misalnya, kita tahu dia pernah aktif sebagai guru di iskandariah, mesir, di sekitar tahun sm, tetapi kapan dia lahir dan kapan dia wafat betulbetul gelap. The extended Euclidean algorithm is particularly useful when a and b are coprime.
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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.
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. The standard Euclidean algorithm proceeds by a succession of Euclidean divisions whose quotients are not used. Only the remainders are kept. For the extended algorithm, the successive quotients are used. Also it means that the algorithm can be done without integer overflow when a and b are representable integers.
The following table shows how the extended Euclidean algorithm proceeds with input and The greatest common divisor is the last non zero entry, 2 in the column "remainder". The computation stops at row 6, because the remainder in it is 0.
Until this point, the proof is the same as that of the classical Euclidean algorithm. As they are coprime, they are, up to their sign the quotients of b and a by their greatest common divisor. The definitions then show that the a , b case reduces to the b , a case. If a and b are two nonzero polynomials, then the extended Euclidean algorithm produces the unique pair of polynomials s , t such that.
A third difference is that, in the polynomial case, the greatest common divisor is defined only up to the multiplication by a non zero constant. There are several ways to define unambiguously a greatest common divisor.
In mathematics, it is common to require that the greatest common divisor be a monic polynomial. To get this, it suffices to divide every element of the output by the leading coefficient of r k. Otherwise, one may get any non-zero constant. In computer algebra , the polynomials commonly have integer coefficients, and this way of normalizing the greatest common divisor introduces too many fractions to be convenient.
If the input polynomials are coprime, this normalisation also provides a greatest common divisor equal to 1. The drawback of this approach is that a lot of fractions should be computed and simplified during the computation. A third approach consists in extending the algorithm of subresultant pseudo-remainder sequences in a way that is similar to the extension of the Euclidean algorithm to the extended Euclidean algorithm.
This allows that, when starting with polynomials with integer coefficients, all polynomials that are computed have integer coefficients. To implement the algorithm that is described above, one should first remark that only the two last values of the indexed variables are needed at each step.
Thus, for saving memory, each indexed variable must be replaced by just two variables. For simplicity, the following algorithm and the other algorithms in this article uses parallel assignments.
In a programming language which does not have this feature, the parallel assignments need to be simulated with an auxiliary variable. For example, the first one,. The quotients of a and b by their greatest common divisor, which is output, may have an incorrect sign. This is easy to correct at the end of the computation but has not been done here for simplifying the code.
Similarly, if either a or b is zero and the other is negative, the greatest common divisor that is output is negative, and all the signs of the output must be changed. When using integers of unbounded size, the time needed for multiplication and division grows quadratically with the size of the integers. This canonical simplified form can be obtained by replacing the three output lines of the preceding pseudo code by.
To get the canonical simplified form, it suffices to move the minus sign for having a positive denominator. It is the only case where the output is an integer. The extended Euclidean algorithm is the essential tool for computing multiplicative inverses in modular structures, typically the modular integers and the algebraic field extensions.
A notable instance of the latter case are the finite fields of non-prime order. In particular, if n is prime , a has a multiplicative inverse if it is not zero modulo n. Thus t , or, more exactly, the remainder of the division of t by n , is the multiplicative inverse of a modulo n.
This results in the pseudocode, in which the input n is an integer larger than 1. The extended Euclidean algorithm is also the main tool for computing multiplicative inverses in simple algebraic field extensions. An important case, widely used in cryptography and coding theory , is that of finite fields of non-prime order. The addition in L is the addition of polynomials. The multiplication in L is the remainder of the Euclidean division by p of the product of polynomials.
Thus, to complete the arithmetic in L , it remains only to define how to compute multiplicative inverses. This is done by the extended Euclidean algorithm. The algorithm is very similar to that provided above for computing the modular multiplicative inverse. In the pseudocode which follows, p is a polynomial of degree greater than one, and a is a polynomial. Moreover, div is an auxiliary function that computes the quotient of the Euclidean division. Since 1 is the only nonzero element of GF 2 , the adjustment in the last line of the pseudocode is not needed.
One can handle the case of more than two numbers iteratively. From Wikipedia, the free encyclopedia. Main article: Modular arithmetic. Number-theoretic algorithms. Binary Euclidean Extended Euclidean Lehmer's. Cipolla Pocklington's Tonelli—Shanks Berlekamp. Categories : Number theoretic algorithms Euclid. Hidden categories: Articles with example pseudocode. Namespaces Article Talk. Views Read Edit View history. Contribute Help Community portal Recent changes Upload file.
Extended GCD Algorithm
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. 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.
Extended Euclidean algorithm