EEM 465
FUNDAMENTALS of DATA COMMUNICATIONS
Lecture 7
Lecturer
Assist.Prof.Dr. Nuray At
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Galois Fields
Group: A group G is a set of objects on which an operation “ . ” has been defined.
Group properties:
1) Closure: If
2) Associativity:
,
3) Identity: There exists
such that
4) Inverse:
, there exists a unique element
such that
A group is said to be commutative if it also satisfies
5)
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Example: Is the set of integers under integer addition a group?
1) Closure? Yes
2) Associativity? Yes
3) Identity? Yes
4) Inverse? Yes
5) Commutative? Yes
Example: Is the set of integers under integer multiplication a group?
1) Closure? Yes
2) Associativity? Yes
3) Identity? Yes. It is 1
4) Inverse? No
Not a group!
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 The order of a group is defined to be the cardinality of the group.
One of the simplest methods for constructing finite groups lies in the application
of modular arithmetic to the set of integers.
Addition mod‐m
Example:
13 + 18 = 11 mod 20
15 + 100 = 50 mod 65
Equivalence Classes:
Addition mod‐m groups the set of integers into m distinct equivalence classes.
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Example: Equivalence classes of integers under mod‐5 addition
Label Equivalence class
Theorem: The equivalence classes { 0, 1, 2,..., m ‐ 1} form a commutative group of
order m under mod‐m integer addition.
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Example: The group of order 4 under mod‐4 addition
Modular Multiplication
Perform usual multiplication and reduce result mod‐m
Example:
11 ∙ 5 = 15 mod 20
6 ∙ 4 = 6 mod 18
 Modular multiplication cannot be used to form a finite group using the
integers and arbitrary moduli.
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Theorem: S = { 1, 2, 3,..., p ‐ 1} forms a commutative group of order p‐1 under
mod‐p multiplication if and only if p is a prime integer.
Example: G ={1 , 2, 3, 4} mod‐5 multiplication
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Definition: Order of a group element
Let g be an element in the group G with group operation “ . ” Let g2 = g ∙ g and
g3 = g ∙ g ∙g. The order of g is the smallest positive integer ord(g) such that gord(g) is
the group identity element, that is, gord(g) = e
Example: G ={1, 2, 3, 4} mod‐5 multiplication
ord(1)=1
2.2.2.2=1 mod 5, ord(2)=4
3.3.3.3=1 mod 5, ord(3)=4
4.4=1
mod 5, ord(4)=2
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Ring: A collection of elements R with two operations “ + ” and “ . ” such that
1) R forms a commutative group under “ + ”. Identity “ 0 ”
2) “ . ” is associative.
3) The operation “ . ” distributes over “ + ”:
A ring is said to be a commutative ring if
4) “ . ” commutes
A ring with identity
5) “ . ” has an identity element “ 1 ”
Example: The integers under mod‐m addition and multiplication form a
commutative ring with identity.
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Field: F is a set of objects on which two operations “ + ” and “ . ” are defined
1) F forms a commutative group under “ + ”. Identity “ 0 ”
2) F ‐ {0} forms a commutative group under “ . ” . Multiplicative identity “ 1 ”
3) “ + ” and “ . ” distribute:
 Fields can be viewed as commutative rings with identity and multiplicative
inverses.
 All of the field elements form an additive commutative group, while the
nonzero elements form a multiplicative commutative group.
Fields of finite order are known as Galois Fields.
A Galois field of order q is usually denoted by GF(q).
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Example: The simplest of the Galois fields is GF(2)
Two element set {0, 1} under standard binary addition and multiplication
Theorem: The integers {0, 1, 2,…, p‐1} where p is a prime, form the field GF(p)
under mod‐p addition and multiplication.
Example: GF(3) = {0, 1, 2} under mod‐3 addition and multiplication
References
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Wicker, S. B., “Error control systems for digital communication and storage”
Prentice‐Hall, 1995.
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