The extended euclidean algorithm

The extended euclidean algorithm

Given two integers a and b, we often need to find other two integers s and t such that


 s * a + t * b = gcd ( a, b )

The extended euclidean algorithm can calculate the gcd ( a, b ) and at the same time calculate the values of s and t. The algorithm uses three sets of variables r's, s's and t's. In each state r1, r2 and r have same value in the euclidean algorithm. The variables r1, r2 are initializes t the value of a and b respectively. The variables s1, s2 are initialized to 1 and 0 respectively. The variables t1 and t2 are initialized t 0 and 1 respectively.



















 r1 = a ; r2 = b ; ( Initialization)
s1 ← 1 ; s2 ← 0 ;
t1 ← 0 ; t2 ← 1 ;
 while ( r2 > 0)
 {

   q = r1 / r2;
   r = r1 - q * r2 ; (updating r )
   r1 ← r2;  r2 ← r ;
   s ← s1 - q * s2 ;
   s1 ← s2 ; s2 ← s ; (updating S's)
   t ← t1 - q * t2 ; 
   t ← t2 ; t2 ← t ; (updating T's)

 }




gcd (a , b ) ← r1 ;   s ← s1 ;   t ← t1 ;

Q. Given a =161, b = 28. Find gcd(a,b) and the values of s and t by using extended euclidean algorithm.




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