5. Suppose that GF (2³) is defined by the irreducible polynomial x³ + x + 1. From 4., deduce that the multiplicative inverse 101-¹ of 101 in GF(2³) is: [Hint: The elements of GF (2³) can be represented by polynomials ax² + bx+c whose coefficients a, b, c are in GF (2), which in this case are the bits 0,1. We can represent the polynomials by the corresponding bit-strings: e.g., x² + 1 → 101 and x + 1 → 011. Then if we reduce he product of the polynomials modulo the irreducible polynomial x³ + x + 1, this will correspond to the product: 010 · 101. What was the product of the polynomials congruent to? The answer should follow.]
5. Suppose that GF (2³) is defined by the irreducible polynomial x³ + x + 1. From 4., deduce that the multiplicative inverse 101-¹ of 101 in GF(2³) is: [Hint: The elements of GF (2³) can be represented by polynomials ax² + bx+c whose coefficients a, b, c are in GF (2), which in this case are the bits 0,1. We can represent the polynomials by the corresponding bit-strings: e.g., x² + 1 → 101 and x + 1 → 011. Then if we reduce he product of the polynomials modulo the irreducible polynomial x³ + x + 1, this will correspond to the product: 010 · 101. What was the product of the polynomials congruent to? The answer should follow.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![5. Suppose that GF (2³) is defined by the irreducible polynomial x³ + x + 1. From 4., deduce that
the multiplicative inverse 101-¹ of 101 in GF(2³) is:
[Hint: The elements of GF (2³) can be represented by polynomials ax² + bx+c whose
coefficients a, b, c are in GF (2), which in this case are the bits 0,1. We can represent the
polynomials by the corresponding bit-strings: e.g., x² + 1 → 101 and x + 1 → 011.
Then if we reduce he product of the polynomials modulo the irreducible polynomial x³ + x +
1, this will correspond to the product: 010 · 101. What was the product of the polynomials
congruent to? The answer should follow.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59cda1e2-bddd-4dda-aa68-0c4c70ff4fb7%2Fa5082657-95bb-4418-9333-702f5c4107d3%2Fn9rj0pe_processed.png&w=3840&q=75)
Transcribed Image Text:5. Suppose that GF (2³) is defined by the irreducible polynomial x³ + x + 1. From 4., deduce that
the multiplicative inverse 101-¹ of 101 in GF(2³) is:
[Hint: The elements of GF (2³) can be represented by polynomials ax² + bx+c whose
coefficients a, b, c are in GF (2), which in this case are the bits 0,1. We can represent the
polynomials by the corresponding bit-strings: e.g., x² + 1 → 101 and x + 1 → 011.
Then if we reduce he product of the polynomials modulo the irreducible polynomial x³ + x +
1, this will correspond to the product: 010 · 101. What was the product of the polynomials
congruent to? The answer should follow.]
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