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43. Galois GF(2^4) Finite Field

 Galois GF(2^4) Finite Field  <24/December/2015>.

Today I am presenting a bit about  Galois GF(2^4) Finite Field constructed under the irreducible polynomial p(x)=x4+x3+1. A root of p(x) is the prime number  b=0010 and we can constuct a normal basis using the powers b8 ,b4 , b2 , b.
You will find useful about the followings the polynomial calculator included in article N0 39.

we have :
b =0010
b2=0100
b3=1000
b4=1001
b5=1011
b6=1111
b7=0111
b8=1110

so, a normal basis for GF(2^4) Finite Field
is the set B={b8, b4 , b2 , b}

The elements of GF(2^4) Finite Field  can be represented now using B.
we have:

Table 1
0--->0000
1----1111
2----0001
3----1110
4----0010
5----1101
6----0011
7----1100
8----1011
9----0100
10---1010
11---0101
12---1001
13---0110
14---1000
15---0111

The elements at left are presented in decimal form. For examle 11=0xB and its normal basis representation is 0x5.

Lets try now to square polynomials !

Example 1

A(x) = 10 x2 + 0 x + 0 
 
B(x) = 10 x2 + 0 x + 0 
 
A(x) × B(x) = 11 x4 + 0 x3 + 0 x2 + 0 x + 0 

Observe that: 10----->normal basis representation=1010---------->one cyclic left shift=0101--------->using table 1 to come back=11.

So, we can find the normal basis representation of the square doing one cyclic left shift simply.

Example 2

A(x) = 6 x2 + 0 x + 0 
 
B(x) = 6 x2 + 0 x + 0 
 
A(x) × B(x) = 13 x4 + 0 x3 + 0 x2 + 0 x + 0 

As in example 1 we have:

6----->0011----->0110------>13

Bibliography:

http://faculty.washington.edu/manisoma/ee540/EE540finite.pdf

http://www.famnit.upr.si/sl/resources/files/seminars/amela-muratovic-ribic.pdf

Mike Rosing :
http://www.embeddedrelated.com/showarticle/867/optimal-normal-basis


 

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