欧几里德算法 欧几里德算法又称辗转相除法,用于计算两个整数a,b的最大公约数。 基本算法:设a=qb+r,其中a,b,q,r都是整数,则gcd(a,b)=gcd(b,r),即gcd(a,b)=gcd(b,a%b)。 第一种证明: a可以表示成a = kb + r,则r = a mod b 假设d是a,b的一个公约数,则有 d|a, d|b,而r = a ...
Example of Extended Euclidean AlgorithmRecall that gcd(84, 33) = gcd(33, 18) = gcd(18, 15) = gcd(15, 3) = gcd(3, 0) = 3 We work backwards to write 3 as a linear combination of 84 and 33: 3 = 18 − 15 [Now 3 is a linear combination of 18 and 15] = 18 − (33...
欧几里德与扩展欧几里德算法 Extended Euclidean algorithm,欧几里德算法欧几里德算法又称辗转相除法,用于计算两个整数a,b的最大公约数。基本算法:设a=qb+r,其中a,b,q,r都是整数,则gcd(a,b)=gcd(b,r),即gcd(a,b)=gcd(b,a%b)。第一种证明:a可以表示成a=kb+r,则r=a
We know that by means of extended euclidean algorithmxandycan be calculated fromax + by = gcd(a, b).The formula is: x=prev_y;y=prev_x-(a/b)*x; and the code is: intgcd(inta,intb,int&x,int&y){if(b==0){x=1;y=0;returna;}intx1,y1;intd=gcd(b,a%b,x1,y1);x=y1;y...
前置芝士:欧几里德算法与裴蜀定理。 我们考虑欧几里德算法的最后一步,当b=0b=0时,要使得 ax+0y=gcd(a,0)=aax+0y=gcd(a,0)=a 成立,那么只要取x=1x=1,yy取任意整数即可,不妨取y=0y=0. 因为gcd(a,b)=gcd(b,amodb)gcd(a,b)=gcd(b,amodb),所以可以考虑当整数x,yx,y使得 ...
3) extended Euclidean 扩展欧几里德算法 1. The modular inverse operation byextended Euclideanalgorithm and the binaryextended Euclideanalgorithm are analyzed, and the improved modular inverse operation by binaryextended Euclideanalgorithm is presented in this paper. ...
Extended euclidean algorithmLarge-scale polynomialsCloud computing gives resource-constrained clients great conveniences to outsource exorbitant computations to a public cloud. The extended Euclidean algorithm with large-scale polynomials over finite fields is fundamental and widespread in computer science and ...
Examples The functionegcdis a pure-Python implementation of theextended Euclidean algorithmthat can be viewed as an expansion of the functionality and interface of the built-inmath.gcdfunction. When it is supplied two integer argumentsaandb, it returns a tuple of the form(g, s, t)where the ...
在中文区讲解这两个算法的博客虽然不少,但是让人看得像是在云里雾里的。 本文通过一个详细的计算过程来讲解辗转相除法(即欧几里得算法),以及拓展的欧几里得算法/Extended Euclidean algorithm。这两个算法有很强的关联,因此请读者先看完辗转相除法再看拓展的欧几里得算法。 辗转相除法(即欧几里得算法) 目的:计算两个...
算法4.1 - 辗转相除法(Euclid's Algorithm) 定理4.2 - 辗转相除法 定理4.3 - 裴蜀定理 证明(扩展欧几里得算法) 练习1 练习2(复数) 推论4.4 - 素数的基本性质 证明 推论4.5 - 普遍的情况 证明 定理4.5 - 算术基本定理(Fundamental Theorem of Arithmetic) ...