L`algorithme d`Euclide étendu

ax +by = 1 (a, b)=1
95x+ 14y= 1
a, b... x, y... ±
95 34 −14 ·34 −95 ·5=1
6 14 5
OO
+ 6 ·5 + 4 = 34 14 ·4−11 ·5 = 1
1 11 4
OO
−2·4 + 1 = 5 3 ·4−11 ·1=1
3 3 1
OO
+ 3 ·1 + 1 = 4 3 ·1−2·1=1
1 2 1
OO
−1·1 + 0 = 1 1 ·1−2·0=1
2 1 0
OO
+ 2 ·0 + 1 = 1 1 ·1−0·0=1
0 1 −
a b →q c
0 1
1
−1
x0=y0·q+z0
ay −bx =±1

3 3 ·4−11 ·1=1
a, b... x, y...
95 34
6 14 5
OO
1 11 4
OO
3 1
OO
+1 −1
l1=l0
1·q+l0 2l2=l0 1.
±1
a x
95 34
95 34
14 5
=−1→14 ·34 −95 ·5=1
6 14 5
14 5
11 4
= +1 →14 ·4−11 ·5 = 1
1 11 4
11 4
3 1
=−1→3·4−11 ·1=1
3 3 1
3 1
2 1
= +1 →3·1−2·1=1
1 2 1
2 1
1 0
=−1→1·1−2·0=1
2 1 0
1 0
0 1
= 1 →1·1−0·0=1
0 1
±1±d
d= (a, b)

[1] ay −bx = 1
a b a =b·q+c c =a−b·q
(b·q+c)y−bx = 1
b(qy −x) + cy = 1
b(x−qy)
| {z }
z
−cy =−1
z=x−qy x =q·y+x
[2] bz −cy =−1
x=q·y+z a =b·q+c
a x
b y
= 1
(a−bq) (x−qy)
b y
= 1 ⇐⇒
c z
b y
= 1 ⇐⇒
b y
c z
=−1