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.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
Related questions
Question
![(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, but
it 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); your
polynomials 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 Euclidean
algorithm for polynomials. Each step in the Euclidean algorithm is a
long division of polynomials with remainder. At every step, just make
sure 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 polynomials
s(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 this
new polynomial, compute again Parts (a) and (c).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F503d0b54-dd3c-4280-b1c1-bad565a93f00%2F4ff604e9-8580-427e-a35c-5cd87f82bea5%2Fsdmadw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(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, but
it 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); your
polynomials 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 Euclidean
algorithm for polynomials. Each step in the Euclidean algorithm is a
long division of polynomials with remainder. At every step, just make
sure 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 polynomials
s(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 this
new polynomial, compute again Parts (a) and (c).
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