Extended Euclidean Algorithm and code implementation

The Extended Euclidean algorithm is:

Ax + by = gcd (A, B)

Returns an integer (x, y)

1. Non-Recursive Implementation:

First, let's look at the situation where a = 60, B = 22:

The left side of the table is the Euclidean algorithm, and the right side of the equation is used to calculate the solutions of AX + by = gcd (A, B ).

A = 2 × B + 16	16 = A-2b
B = 1 × 16 + 6	6 = B-1 × 16 = B-1 × (a-2b) =-A + 3B
16 = 2x6 + 4	4 = 16-2 × 6 = (A-2b)-2 × (-A + 3b) 3A-8b
6 = 1X4 + 2	2 = 6-1X4 = (-A + 3b)-1 × (3A-8b) =-4A + 11b
4 = 2X2 + 0

 

 

 

 

 

 

 

 

 

 

 

This is the pseudocode in number theory guidance:

1. Set x = 1, G = A, V = 0 and W = B
2. If W = 0, set Y = (G-Ax)/B and return values (G, x, y)
3. G divide by W to get the remainder T, G = QW + T
4. Set S = x-QV
5. Set (x, G) = (V, W)
6. Set (V, W) = (S, T)
7. Go to step 2

So how is X calculated? X is the coefficient of A in the last remainder minus the obtained Q multiplied by a in the last remainder.

That is, R (n) = R (n-2)-Q (n) × R (n-1) (The number in parentheses represents the subscript)

This sentence is a bit round, = _ = !!..

Let's look at the general table:

A = q1b + r1	R1 = A-q1b
B = q2r1 + R2	R2 = B-q2r1
R1 = q3r2 + r3	R3 = R1-q3r2
......	......

 

 

 

 

 1 int ex_gcd1(int a, int b, int &x, int &y) 2 { 3     int g, v, w, s, t, q; 4     x = 1; 5     v = 0; 6     g = a; 7     w = b; 8     while(w != 0) 9     {10         q = g / w;11         t = g % w;12         s = x - q*v;13         x = v;14         g = w;15         v = s;16         w = t;17     }18     y = (g - a*x) / b;19     return g;20 }

Code Jun

 

II. Implementation of Recursion

Make a' = A % B, T = A/B, y' = Y + Tx

Ax + by = G

Ax-tbx + by = G

(A-TB) x + B (Tx + y) = g

A' x + by' = G

By '+ a' x = G

Y = y'-TX, y' is a recursive y that is deeper than Y

 1 int ex_gcd2(int a, int b, int &x, int &y) 2 { 3     if(b == 0) 4     { 5         x = 1; 6         y = 0; 7         return a; 8     } 9     int g = ex_gcd2(b, a%b, y, x);10     y = y - a/b*x;11     return g;12 }

Code Jun