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I am trying to find the inverse of $x ^3+x +1$ in $GF(8)$ defined on the quotient ring $GF(2)[x]/\langle x^3+x+1\rangle$. I have done the Euclidean algorithm but I am stuck in the forward process to get the inverse. Please explain how to do it from reverse.

Bingming
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Mariam10
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  • Do it this way - much less painful and less error-prone. – Bill Dubuque Nov 15 '18 at 00:25
  • Normally $x$ stands for an indeterminate, when $x^3+x+1$ is a polynomial. Polynomials don't have inverses in a finite field. If, as I suspect, $x$ stands for some element of $GF(8)$ you need to tell us which element. Or at least, tell us its minimal polynomial. You see, very often the field $GF(8)$ is defined as the quotient ring $GF(2)[x]/\langle x^3+x+1\rangle$. But, in that field the coset of $x^3+x+1$ is equal to zero. Again implying that it has no inverse (can't divide by zero). – Jyrki Lahtonen Nov 15 '18 at 04:14
  • The (only) other alternative is to have defined $GF(8)$ as the quotien $GF(2)[x]/\langle x^3+x^2+1\rangle$. In that case the coset of $x^3+x+1$ equals the coset of $x^2+x$ (because $(x^3+x+1)-(x^3+x^2+1)=x^2+x$). You can find the inverse of that by the usual process involving extended Euclid. You see, to positively id an element of the quotient ring we customarily reduce them by the polynomial defining the field. – Jyrki Lahtonen Nov 15 '18 at 04:17
  • By saying finite field, i mean the polynomial of the field, which is x^8+ x^4+ x^3+x+1. So i have to find the inverse of the first polynomial modulus this one – Mariam10 Nov 15 '18 at 07:20
  • What do you mean by $x^8 + x^4 + x^3 + x + 1$ being the "polynomial of the field"? – Tobias Kildetoft Nov 15 '18 at 09:02
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    But, Mariam, the polynomial $x^8+x^4+x^3+x+1$ defines the version of the field $GF(2^8)=GF(256)=GF(2)[x]/\langle x^8+x^4+x^3+x+1\rangle$ IIRC used in AES cryptosystem (among other things). – Jyrki Lahtonen Nov 15 '18 at 20:10
  • For finding inverses in the AES field with Extended Euclid I recommend that you take a peek at this old thread. Robjohn spelled it out! – Jyrki Lahtonen Nov 15 '18 at 20:12
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    Here I spell out what $GF(8)$, also known as $\Bbb{F}_8$, looks like. If I were to use $GF(256)$ in a computer program I would build a similar logarithm table for $GF(256)$ and use it calculate inverses. – Jyrki Lahtonen Nov 15 '18 at 20:15

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Hint: The field $GF(8)$ is isomorphic to $GF(2)[x]/\langle x^3+x+1\rangle$ or to $GF(2)[x]/\langle x^3+x^2+1\rangle$. The polynomials $x^3+x+1$ and $x^3+x^2+1$ are the only irreducible polynomials over $GF(2)$ (and they are conjugate). The elements of $GF(8)$ are the residue classes of the polynomials $ax^2+bx+c$, where $a,b,c\in GF(2)$. So maybe you should rethink about your question.

Wuestenfux
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