Question

determine the multiplicative inverse of x^3 + x + 1 in gf(2^4) with m(x) = x4 + x + 1

Answer 1

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Answer 2

Given:

• $x^{3} + x + 1$

• $GF(2^{4} )$

• m(x) = $x^{4} + x + 1$ (This is a primitive polynomial)

To find :

• The Multiplicative Inverse.

Solution:

• GF is called  the Galois Field which can be written in the form $F_(2^{4})$
• Here $F_(2^{4})$ is the quotient ring of $x^{4} = X + 1$
• The above equation is in the field of $2^{4}$
• We get elements of $GF(2^{4} )$ by defining primitive polynomial,
• $a_3x^{3} +a_2x^{2} +a_1x+a_0$
• But $x^{4}$ is not an element of the field, we can reduce it by primitive polynomial.
• But also we will get the inverse as the same.

The Multiplicative Inverse is the same.