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(a). Show that the addition of x²+x and x²+x+1 is in GF(2³) (b).Show that the multiplication of x²+x and x²+x+1 is in GF(2³)

(a). Show that the addition of x²+x and x²+x+1 is in GF(2³) (b).Show that the multiplication of x²+x and x²+x+1 is in GF(2³)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a). Show that the addition of x2+x and x²+x+1 is in GF(2³) (b).Show that the multiplication of x²+x and x2+x+1 is in GF(2³)
Transcribed Image Text:(a). Show that the addition of x2+x and x²+x+1 is in GF(2³) (b).Show that the multiplication of x²+x and x2+x+1 is in GF(2³)
Expert Solution
Step 1

We know that GF(2m) are the binary polynomials, i.e. polynomial whose coefficients are either 0 or 1.

Here given GF(23). there are 23 polynomials in the field and the degree of each polynomials is no more than 3-1=2.

So

GF(23)=0, 1, x, x+1, x2, x2+1,x2+x, x2+x+1 .

Solution for a:

Addition of x2+x and  x2+x+1 is

(x2+x)+(x2+x+1)=2x2+2x+1

Since coefficients are either 0 or 1.

So after reduction modulo 2, we get

(x2+x)+(x2+x+1)=0x2+0x+1⇒(x2+x)+(x2+x+1)=1

Hence addition of x2+x and  x2+x+1 is in GF(23).

 

 

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