1 Groups, Rings and Field:
2 Modular Arithmetic

**FINITE FIELDS**

**1 Groups, Rings and Field:**

**Group**: A set of elements that is
closed with respect to some operation.

Closed->
The result of the operation is also in the set

The
operation obeys:

Obeys associative law: **(a.b).c =
a.(b.c)**^{}

^{ }

Has identity e: **e.a = a.e = a**^{}

^{ }

Has inverses **a-1: a.a-1 = e**^{}

^{ }

**Abelian Group**: The operation is commutative

**a.b = b.a**

Example:
Z8, + modular addition, identity =0

**Cyclic Group**

Exponentiation:
Repeated application of operator

example: **a3 = a.a.a**

Cyclic
Group: Every element is a power of some fixed element, i.e., **b = ak** for some a and every b in group
a is said to be a generator of the group

Example:
{1, 2, 4, 8} with mod 12 multiplication, the generator is 2.

20=1,
21=2, 22=4, 23=8, 24=4, 25=8

**Ring:**

A group
with two operations: addition and multiplication

The group
is abelian with respect to addition: **a+b=b+a**

Multiplication
and additions are both associative:

**a+(b+c)=(a+b)+c**

**a.(b.c)=(a.b).c**

Multiplication
distributes over addition, **a.(b+c)=a.b+a.c**

Commutative
Ring: Multiplication is commutative, i.e., **a.b
= b.a**

Integral
Domain: Multiplication operation has an identity and no zero divisors

**Field:**

An
integral domain in which each element has a multiplicative inverse.

**2 Modular Arithmetic**

modular
arithmetic is 'clock arithmetic'

a **congruence** a = b mod n says when
divided by *n* that a and b have the
same remainder o 100 = 34 mod 11

usually
have 0<=b<=n-1

-12mod7 =
-5mod7 = 2mod7 = 9mod7 o b is called the **residue** of a mod n

can do
arithmetic with integers modulo n with all results between 0 and n

**Addition**

**a+b mod n**

**Subtraction**

**a-b mod n = a+(-b) mod n**

**Multiplication**

**a.b mod n**

derived
from repeated addition

can get
a.b=0 where neither a,b=0 o eg 2.5 mod 10

**Division**

**a/b mod n**

is
multiplication by inverse of b: a/b = a.b^{-1} mod n

if n is
prime b^{-1} mod n exists s.t b.b^{-1} = 1 mod n o eg 2.3=1 mod 5 hence 4/2=4.3=2
mod 5

integers
modulo n with addition and multiplication form a commutative ring with the laws
of

**Associativity **: (a+b)+c = a+(b+c) mod n

**Commutativity **: a+b = b+a mod n

**Distributivity **: (a+b).c = (a.c)+(b.c) mod n

also can
chose whether to do an operation and then reduce modulo n, or reduce then do
the operation, since reduction is a homomorphism from the ring of integers to
the ring of integers modulo n

a+/-b mod
n = [a mod n +/- b mod n] mod n o (the above laws also hold for multiplication)

if n is
constrained to be a prime number *p*
then this forms a **Galois Field modulo p**

denoted **GF(p)** and all the normal laws
associated with integer arithmetic work

**Greatest Common Divisor**

the
greatest common divisor (a,b) of a and b is the largest number that divides
evenly into both a and b

**Euclid's Algorithm **is used
to find the Greatest Common Divisor (GCD) of two numbers** **a and n, a<n

use fact
if a and b have divisor d so does a-b, a-2b GCD (a,n) is given by:

let g0=n
g1=a

gi+1 =
gi-1 mod gi when gi=0 then (a,n) = gi-1

eg find
(56,98) g0=98 g1=56

g2 = 98
mod 56 = 42

g3 = 56
mod 42 = 14

g4 = 42
mod 14 = 0 hence (56,98)=14

**Finite Fields or Galois Fields**

Finite
Field: A field with finite number of elements

Also
known as Galois Field

The
number of elements is always a power of a prime number. Hence, denoted as
GF(pn)

GF(p) is
the set of integers {0,1, …, p-1} with arithmetic operations modulo prime p

Can do
addition, subtraction, multiplication, and division without leaving the field
GF(p)

GF(2) =
Mod 2 arithmetic GF(8) = Mod 8 arithmetic

There is
no GF(6) since 6 is not a power of a prime

**Polynomial Arithmetic**

*f***( x) = a_{n}x^{n} + a_{n-1}x^{n-1} + …+ a_{1}x + a_{0} = Σ a_{i}x^{i}**

Ordinary
polynomial arithmetic:

Add, subtract, multiply, divide polynomials,

Find remainders, quotient.

Some polynomials have no factors and are prime.

Polynomial
arithmetic with mod p coefficients

Polynomial
arithmetic with mod p coefficients and mod m(x) operations

**Polynomial Arithmetic with Mod 2 Coefficients**

All
coefficients are 0 or 1, e.g.,

let *f*(*x*)
= *x*3 + *x*2 and *g*(*x*) = *x*2
+ *x* + 1 *f*(*x*) +* g*(*x*)
=* x*3 +* x *+ 1

*f*(*x*) x*
g*(*x*) =* x*5 +* x*2

Polynomial
Division: *f*(*x*) = *q*(*x*) *g*(*x*) + *r*(*x*)

can
interpret *r*(*x*) as being a remainder

*r*(*x*) =* f*(*x*) mod* g*(*x*)

if no
remainder, say *g*(*x*) divides *f*(*x*)

if *g*(*x*)
has no divisors other than itself & 1 say it is irreducible (or prime)
polynomial

Arithmetic
modulo an irreducible polynomial forms a finite field

Can use
Euclid‟s algorithm to find gcd and inverses.

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