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I. Finite Field Algebra
Binary Operation
G is a set of elements
a , b G
c  a* b
if c  G
© Tallal Elshabrawy
“*” A binary operation
on G is a rule that
assigns to each pair of
elements a and b a
uniquely defined
element c
G is closed under “*”
2
Groups
A set G on which a binary operation “*” is
defined is called a Group if:
i. The binary operation is associative
ii. G contains an identity element e
(a *e = e *a = a)
iii. For any element a in G, there exists an
inverse element a’ in G
(a *a’ = a’ *a = e)
Commutative Group G if for any a and b in G:
a*b = b*a
© Tallal Elshabrawy
3
Theorems
The identity element in a group G is unique
Proof If we have two identity elements e and
e’ in G, Then,
e’ =e’ * e =e  e, e’ are identical
The inverse of any element in a group G is unique
Proof If we have two inverse elements a’ and
a’’ for a in G, Then,
a’ =a’ *e =a’ *(a*a’’)  a’, a’’ are
identical
© Tallal Elshabrawy
4
Example: Modulo-2 Addition
The set G={0,1} is a group of order 2 under
modulo-2 addition
i. Modulo-2 addition is
associative
ii. The identity element is 0
iii. The inverse of 0 is 0 in G
The inverse of 1 is 1 in G
Modulo-2
Addition
00  0
01  1
1 0  1
11  0
© Tallal Elshabrawy
5
Example: Modulo-m Addition
The set G={0,1,2,…,m-1} is a group of order m
under modulo-m addition
i. Modulo-m addition
is associative
ii. The identity
element is 0
iii. The inverse of i is
m-i in G
© Tallal Elshabrawy
Modulo-m
Addition
i
+ j =r
i+j=qm+r,
0≤r<m-1
6
Example: Modulo-p Multiplication
G={1,2,…,p-1}, p is a
prime number, is a
group of order p under
modulo-p multiplication
Modulo-5 multiplication is
associative
ii. The identity element is 1
iii. The inverse of 1 is 1 in G
The inverse of 2 is 3 in G
The inverse of 3 is 2 in G
The inverse of 4 is 4 in G
i.
© Tallal Elshabrawy
Modulo-p
Multiplication
. j =r
i.j=qp+r, 0≤r<p-1
i
Modulo-5 Multiplication
. 1
.
1 1
2 2
2
2
4
3
3
1
4
4
3
3
3
1
4
2
4
4
3
2
1
7
SubGroups
Define a set G as a group under a binary
operation *, A subset H is called a subgroup if
i. H is closed under the binary operation *
ii. For any element a in H, the inverse of a is
also in H
Example:
Let G be the set of rational numbers constitute a group
under real addition. Therefore,
The set of integers H is a proper (i.e., H ≠G)
subgroup under real addition
© Tallal Elshabrawy
8
Cosets
H is a subgroup of a group G under binary
operation *
a G
a* H
H* a
a * h : h  H 
h * a : h  H 
Left Coset of H
Right Coset of H
If the group G is commutative,
a *H =H *a is simply labeled as:
a Coset of H
© Tallal Elshabrawy
9
Example
 G={0,1,2,…,15} under modulo-16 addition
 H={0,4,8,12} is a subgroup of G why?
 The coset 3 + H ={3,7,11,15}= 7
+ H
Four Distinct and Disjoint Cosets of H
0
1
2
3
+
+
+
+
© Tallal Elshabrawy
H
H
H
H
={0,4,8,12}
={1,5,9,13}
={2,6,10,14}
={3,7,11,15}
10
Theorem (Read Only)
Let H be a subgroup of a group G with binary
operation *.
No two elements in a Coset of H are identical
a G
a *H
a *h : h  H 
Suppose a *h , a *h' are identical where h  h'
Given a -1 denotes the inverse of a , then
a 1 *  a *h   a 1 *  a *h' 
e *h  e *h'
h  h' (Contradiction)
© Tallal Elshabrawy
11
Theorem (Read Only)
No two
elements in two
different Cosets
of a subgroup
H of a group G
are identical
a *H  b *H , a , b  G
Suppose a *h  a *H , b *h'  b *H
If a *h =b *h'
 a *h  *h 1   b *h'  *h 1
a  b *h'', h''  h' *h 1
a *H   b *h''  *H
a *H   b *h''  *h : h  H 
a *H   b *h'''  : h'''  H 
a *H  b *H Contradiction
© Tallal Elshabrawy
12
Properties of Cosets
i. Every element in G appears in one
and only one of distinct Cosets of H
ii. All the distinct Cosets of H are disjoint
iii. The union of all distinct Cosets of H
forms the group G
© Tallal Elshabrawy
13
Fields
Let F be a set of elements on which two binary
operations called addition “+” and
multiplication “.” are defined. The set F and
the two binary operations represent a field if:
i.
F is a commutative group under addition. The
identity element with respect to addition is called
the zero element (denoted by 0)
ii. The set of nonzero elements in F is a commutative
group under multiplication. The identity element
with respect to multiplication is called the unit
element (denoted the 1 element)
iii. Multiplication is distributive over addition:
a.(b+c) = a.b + a.c, a, b, c in F
© Tallal Elshabrawy
14
Basic Properties of Fields
 a.0=0.a=0
 If a,b≠0, a.b≠0
 a.b=0 and a≠0 imply that b=0
 -(a.b)=(-a).b=a.(-b)
 If a≠0, a.b=a.c imply that b=c
© Tallal Elshabrawy
15
Binary Field GF(2)
Modulo-2 Addition
Modulo-2 Multiplication
+
0
1
.
0
1
0
0
1
0
0
0
1
1
0
1
0
1
F={0,1} is a Finite field of order 2 under
modulo-2 addition and modulo-2 multiplication
Galois Field of the order 2
© Tallal Elshabrawy
16
Subtraction and Division (GF(7))
Modulo-7 Addition
Modulo-7 Multiplication
+
0
1
2
3
4
5
6
.
0
0
1
2
3
4
5
6
1
1
2
3
4
5
6
2
2
3
4
5
6
3
3
4
5
6
4
4
5
6
5
5
6
6
6
0
2
3
4
5
6
0 0
0 0
0
0 0
0
0
1
0
1
2
3
4
5
6
0
1
2
0
2
4
6
1
3
5
0
1
2
3
0
3
6
2
5
1
4
0
1
2
3
4 0
4 1
5
2 6
3
0
1
2
3
4
5
5
3
1
6
4
2
1
2
3
4
5
6 0
6 5
4
3 2
1
Ex: 3-6=3+(-6)=3+1=4
© Tallal Elshabrawy
0
0
1
Ex: 3/2=3.2-1 =3.4=5
17
Characteristic of a Finite Field GF(q) (Read)
1
2
k
1
1
1
1  1 ,  1  1  1 , ... 1  1  1  ...  1  k

i
i
i
times 
Closed Field  There Exists 2 positive integers m,n, m<n where
n
m
1
1
n m
1  1   1  0

i
i
i
1
Characteristic of the field
Smallest integer  that satisfies

10

i
1
© Tallal Elshabrawy
18
Theorem (Read Only)
Characteristic  of a finite field is prime
Proof
Suppose  is not a prime number such that   km
 k  m 
1   1 .  1  0

i 1
 i 1   i 1 
 k 
m 
  1   0 , or   1   0
 i 1 
 i 1 
This contradicts the assumption that  is
km
  
the smallest integer satisfying   1   0
 i 1 
© Tallal Elshabrawy
19
The order of a Field Element (Read)
a 1  a , a 2  a.a , a 3  a.a.a , ...
non zero elements in GF(q)
Closed Field  There Exists 2 positive integers k,m, m>k
where
a k  a m  a  k .a k  a m k  1
Order of a field Element a
Smallest integer n that satisfies
n
a 1
© Tallal Elshabrawy
20
Theorem (Read Only)
Let a be a nonzero element of a finite field GF(q).
Then aq-1=1
Proof
Let b1 , b2 , ..., bq- 1 be (q -1) non-zero distinct elements of GF(q )
If a is an element in GF(q )

  a.b1  ,  a.b2  , ..., a.bq- 1

represent q-1 distinct nonzero elements


  a.b1  ,  a.b2  , ..., a.bq- 1  b1 .b2 .....bq- 1


 a q- 1b1 .b2 .....bq- 1  b1 .b2 .....bq- 1
 a q- 1  1
© Tallal Elshabrawy
21
Theorem (Read Only)
Let a be a nonzero element in a finite field GF(q).
Let n be the order of a. Then n divides q-1
Proof
Suppose n does not divide q -1
 q - 1  kn  r , 0<r<n
 a kn r  1
kn
r
 a .a  1
This is impossible because 0<r <n and
n is the smallest integer such that a n  1
© Tallal Elshabrawy
22
A Primitive Element of GF(q) (Read)
 A nonzero element a is said to be primitive if the
order of a is q-1
 Example: GF(7)
31=3
41=4
32=2
42=2
33=6
43=1
34=4
35=5
36=1
Order of element 3 is 6
Element 3 is a primitive
element of GF(7)
© Tallal Elshabrawy
Order of element 4 is 3
which is a factor of 6
Element 4 is not a primitive
element of GF(7)
23
Binary Field Arithmetic
Polynomial of degree n over GF(2)
f  X   f 0  f 1 X  f 2 X 2  ...  f n X n ,
f 0 , f 1 , f 2 , ..., f n are in GF(2)={0,1}
fn  1
Polynomials of
Degree 1 over
GF(2)
Polynomials of
Degree 2 over
GF(2)
Polynomials of
Degree n over
GF(2)
X
X2
1+X
1+X2
2n Polynomials
over GF(2) with
degree n
X+X2
1+X+X2
© Tallal Elshabrawy
24
Addition of Two Polynomials over GF(2)
Example:
 g(X) = 1+X+X3+X5
 f(X) = 1+X2+X3+X4+X7
 g(X)+f(X) = X+X2+X4+X5+X7
© Tallal Elshabrawy
25
Division of Two Polynomials over GF(2)
X3
X 3  X 1 X 6
X6
 X 2 (Quotient q(X))
X 5 X 4
X 4 X 3
X5
X 3
X5
X 3 X 2
X2
X
1
X
1
X
1
(Remainder r(X))
© Tallal Elshabrawy
26
Irreducible Polynomials
A polynomial p(X) over GF(2) of degree m is said
to be irreducible over GF(2) if p(X) is not divisible
by any polynomial over GF(2) of degree less than
m but greater than 0
© Tallal Elshabrawy
27
Theorem
Any irreducible polynomial over GF(2) divides
Xn+1
where n=2m-1 and m is the degree of the
polynomial
© Tallal Elshabrawy
28
Primitive Polynomials
 An irreducible polynomial p(X) of degree m is said
to be primitive if the smallest positive integer n for
which p(X) divides Xn+1 is n=2m-1
 Example
 p(X)=X4+X+1 divides X15+1 but does not divide any
Xn+1 for 1≤n<15 (Primitive)
 p(X)= X4+X3+X2+X+1 divides X5+1 (Irreducible but
Not Primitive)
© Tallal Elshabrawy
29
Useful Property of Polynomials over GF(2)
f
2
 X   f 0  f 1 X
 f 2 X 2  ...  f n X n 

2

f  X   f 0  f 1 X  f 2 X 2  ...  f n X n 
2
  f 2
 0
 f f X  f X 2  ...  f X n
0
1
2
n
2

f X  
 f f X  f X 2  ...  f X n
2
n
 0 1

2
n 2

f
X

f
X

...

f
X
1
2
n







2









2
f  X    f 0   f 1 X  f 2 X 2  ...  f n X n

2
f
2
f
2
f
2
© Tallal Elshabrawy
 X    f 0 
2
 f 1X

2

 f2X 2
 X   f 0  f 1 X 2  f 2  X 2 
X   f X 2 
2

2

2



 ...  f n X n

 ...  f n X n

2

2


 where f .f  f
i
i
i

30