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I am trying to understand the AES-256 encryption algorithm as it would be implemented on a gated quantum computer (actually, a simulator), and I am having some trouble understanding the theory behind it. The papers I read start with the ring of polynomials given by $F_2[x]/(1 + x + x^3 + x^6 + x^8)$. What is the significance of the polynomial $1 + x + x^3 + x^6 + x^8$? And how does this relate to $GF(2^8)$?

poncho
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Robert Singleton
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1 Answers1

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To answer the specific question, $F_2[x]/(1 + x + x^3 + x^6 + x^8)$ is isomorphic to $GF(2^8)$. See here for more info.

The polynomial $g(x) = 1 + x + x^3 + x^6 + x^8$ is irreducible over $F_2$, so the quotient is a field. The degree of the polynomial is 8, so it is a degree 8 algebraic extension of $F_2$. In other words, it is $F_{2^8}$.

Elements in $F_2[x]/(g(x))$ are equivalence classes of polynomials modulo $g(x)$.

This is a standard way to construct finite-degree algebraic field extensions.

By the way, I think AES actually has $x^4$ instead of $x^6$ in the polynomial. Not sure if that was a typo in your question or if you read it somewhere.

meshcollider
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