9. Euclid’s Algorithm

9. Euclid’s Algorithm
Euclid’s algorithm is a technique for finding the greatest common
divisor ( a , b)
of two integers or polynomials a and b .
Proposition (2.18, page 48)
Let a and b be two positive integers (or polynomials)
Then if a = q1b
+ r1
for
0 ≤ r1 < b ( 0 ≤ deg( r1 ) ≤ deg(
b)
)
One has a , b)
= ( b , r )
(
1
where ( a , b)
denotes greatest common divisor.
b = q2r1
+ r
M
rn-2
= q r
r = q
n-1
2
+ r
⇒
( b , r ) = ( r
n n-1 n
n+ 1rn
⇒ ( a , b)
1
= r
n
1
, r )
2
Example:
Suppose a = 186, b = 66 ,
then
186 = 66 ∗ 2
66 = 54 ∗1
54 = 12 ∗4
12 = 6 ∗ 2
+ 54
+ 12
+ 6
+ 0
the greatest common divisor is 6 .

Euclid’s Division Algorithm for Polynomials
Given two polynomials a(x)
and b(x)
Their greatest common divisor can be computed by an iterative
application of the division algorithm. If the degree of a(x)
is
greater than the degree of b (x)
, the computation of GCD
( a(
x)
, b(
x)
)
is
a(
x)
= q
b(
x)
= q
( x)
( x)
⋅ b(
x)
+ r
r1 ( x)
= q3
( x)
⋅ r2
( x)
+ r
M
r
n-1
( x)
= q
1
2
n+
1
( x)
⋅ r
⋅ r ( x)
+ r
1
n
( x)
1
2
3
( x)
( x)
( x)
where the iterative process stops when a remainder of zero is
obtained.
Then the greatest common divisor of a(x)
and b(x)
is
r n
( x)
= GCD( a(
x)
, b(
x)
)
Example:
a(
x)
=
b(
x)
=
x
x
3
2
+ 1
+ 1
x
x
3
2
2
+ 1 = ( x + 1)
⋅ x + ( x + 1)
+ 1 = ( x + 1)
⋅ x
∴GCD of a ( x)
and b(
x)
is x + 1

m
10. Arithmetic Operations in GF( 2 )
Primitive Elements
m
Consider the Galois field GF( 2 ) generated by the primitive
polynomial
p ( x)
+
2
m-1 m
= p0 + p1x
+ p2
x + L+
pm−1x
x
Definition:
The element α (a root of p (x)
) whose powers
m
generate all the non-zero elements of GF( 2 )iscalleda
m
primitive element of GF( 2 ).
In fact, any element β
m
in GF( 2 ) whose powers generate all
m
the nonzero elements of GF( 2 ) is a primitive element.
Example:
α
4
and
7
α
4
are also primitive elements of GF( 2 ).

Minimum Polynomial
m
(1) Consider the Galois field GF( 2 ) generated by a primitive
polynomial p(x)
of degree m. Let β be a non-zero
m
element of GF( 2 )
Consider the powers
β,
β
2
,
β
2
2
,
2
L , β ,
i
L
If e is the smallest nonnegative integer for which
2
β e
= β
Then the integer “ e ” is called the exponent of β .
(2) consider the product,
φ(
x)
= ( x + β )( x + β
= a + a x + a x
0
1
2
e−1
2
2
) L ( x + β )
2
e-1
+ L+
ae-1x
+ x
e
is a polynomial of e degree.
We can see that φ(x)
is binary and irreducible over
GF( 2 ). φ(x)
is called the minimal polynomial of the
element β .