(mod(4) This question deals with the finite field GF(28), which can be obtained as Z₂[x]x³+x²+x³+x+1). This is the polynomial the AES cryptosystem uses, butit is not the only polynomial that gives GF(28) (see the last part of this question).In this question, you will learn the extended Euclidean algorithm for polynomials.A brief description of this algorithm is given below.Compute the following elements of Z₂[x] (mod x³ + x² + x³ + x + 1); yourpolynomials must be polynomials of degree at most 7.(a) (x² + x³ + x² + x²) (x³ + x + 1).(b) (x³ + x² + 1) + (x³ + x²)(c) (x² + x + 1) − (x² + x + 1).(d) (x² + x)(x³ + 1)−¹ (I could have also written this as (x² + x)/(x³ + 1)).To compute (x³ + 1)−¹ in GF(28), we proceed as in the case of integers.Step 1. First compute the gcd(x³+1, x³+x²¹+x³+x+1). Here use the Euclideanalgorithm for polynomials. Each step in the Euclidean algorithm is along division of polynomials with remainder. At every step, just makesure that the remainder has a degree less than that of the divisor.Step 2. As in the case of integers, trace your steps back and find polynomialss(x) and t(x) such that(x³ + 1)s(x) + (x³ + x² + x³ + x + 1)t(x) = gcd(x³ + 1, x³ + x² + x³ + x + 1).(e) GF(28) can also be obtained as Z₂[x] (mod x³ + x² + x6 +x+1). Under thisnew polynomial, compute again Parts (a) and (c).

the solution is an given below :

is not the only polynomial that gives GF(28) (see the last part of this question). n this question, you will learn the extended Euclidean algorithm for polynomials. A brief description of this algorithm is given below. Compute the following elements of Z₂[x] (mod x8 + x² + x³ + x + 1); your olynomials must be polynomials of degree at most 7.

is not the only polynomial that gives GF(28) (see the last part of this question). n this question, you will learn the extended Euclidean algorithm for polynomials. A brief description of this algorithm is given below. Compute the following elements of Z₂[x] (mod x8 + x² + x³ + x + 1); your olynomials must be polynomials of degree at most 7.

C++ for Engineers and Scientists

4th Edition

ISBN:9781133187844

Author:Bronson, Gary J.

Publisher:Bronson, Gary J.

Chapter4: Selection Structures

Section: Chapter Questions

Problem 14PP

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