itc lecture 8_new

8/2/2019 ITC Lecture 8_new

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Lecture 8

Number theory and Modular

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Motivation

Error-control codes use check equations.These equations require arithmeticoperations defined for codeword

symbols from finite fields. Finite fields.

Other applications of modular arithmetic:

pseu o-ran om num er genera on public-key cryptography

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Groups

A group G is a set with a binary operation *which together satisfy:

closure: a; bG means c= a * bG:

associativity: In G;(a * b)* c= a *(b * c): identity: G contains an element i such that a=

a * i. (i is 0 for additive group)

nverses: or every a , ere ex s s a- such that a * a-1= i. ( -a for additive group)

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Groups-contd

Definition : If a * b= b *a, we say that thegroup operation iscommutative and that G is acommutative orAbelian group.Examples of Groups:

the integers Zunder addition; e n egers un er a on mo u o p pr me Set of all rational numbers excluding zero is a

commutative group under real multiplication.

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Groups-contd

The no. of elements in a Group-order Finite order- finite Group.

e se = , oge er w + s aCommutative Group.

= - , ,.prime. This set forms a commutative groupunder modulo-p addition and the set of non-

zero e emen s orms a commu a ve groupunder modulo-p multiplication.

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The Subgroup

Let Gbe a group with operation * and H

G. H is a subgroup of G if it is a group undere

operation *emma: s a su group o

His closedunder *

con a ns e nverseo every e emen oEx.: H= {Even integers} is a subgroup of

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Contd

Definition : hj

h * h * h h where* is the group operation.

Lemma: If hG; a finite group, then H3

= {h ,h2

,h3

,..} is a subgroup of G.

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Cyc l ic subgroup

The number c of elements in H is calledorder of the element h.

The set of elements h, ,h2 ,h3,..hc=1 is

called a c cle. A c cle is a sub rou . A group that consists of all powers of

one of its elements is a c clic sub rou .

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Groups-contd..

The integers {0, 1, 2, , m-1} withmodulo-m addition.

The integers {1, 2, ,p-1} with modulo-p

.

Note: If p is non-prime, the set cannot form a

g ou un e mu ca on mu ca veinverse does not exist)

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Rings

A set together with operations + and . isa Ring(R ,+ ,. ) if the following hold :

s a ommu a ve roup un er a on.

Closure property is satisfied. . = . .

Multiplication is distributive over addition.

= . . The identity element for the addition is called 0,

and the identity element for the

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multiplication is called 1.


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Ring:Examples

The set of all real numbers under the usualaddition and multiplication.

multiplication.

The set of all n x n matrices with real valuedelements under matrix addition and matrixmultiplication is a non commutative ring.

e se o a po ynom a s n x w rea va uecoefficients under polynomial addition andol nomial multi lication.

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Fields

Let F be a set of elements with binary additionand multiplication.F is a field iff:

F is a commutative group under addition .

The set of non zero elements in F is aommu a ve group un er mu p ca on.

Multiplication is distributive under addition.

ng s a e every non-zero e emenof R has a multiplicative inverse.

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Contd..

Examples: Q , R and C are fields.(Q-the set of rational numbers;R-the set of realnumbers;C- the set of complex numbers)

Order of a Field:The no. of elements in a field .

Finite Field: Field with finite elements.

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Fini t e Fie ld

Consider the Set G ={0,1,.p-1} where p isprime. This set forms a commutative group- -

zero elements forms a commutative group

under modulo-p multiplication. Modulo-p multiplication is distributive over

modulo-p addition.

modulo-p addition and multiplication.

For =2, we obtain binar field GF 2 .

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For every prime number p, the integers(0, 1,., p) with modulo p arithmeticform a finite field with p elements. This

field is called GF(p).

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Subfields

Asu iel isasu setofafieldwhichitselfisafieldundertheinheritedoperations.The original field is said to be an extension

of the subfield.

Q (rationals) is a subfield of R (reals)

.R is a subfield of C (complex).

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The greatest common divisor of two integers mand n is the largest number that divides both m

.

Two different integers m and n are relatively

prime or co-prime if they have no commonproper divisors.

If m and n are relatively prime, their greatest

.

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Constructing finite fields from Z

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