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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
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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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.]
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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