After reading

NIST may not have you in mind, I decided that a new ISA should support binary Galois Field multiplication. Binary Galois Field is a special case of binary polynomial \( \mathbb{Z}_2[x] \) where addition and subtraction are already supported as XOR, written as \( \oplus \). What is missing for GF is multiplication and division in \( \mathbb{Z}_2[x] \). The binary polynomials are expressed as bit vectors where the \(n\)-th bit is the coefficient of the \(x^n\) term. Let \( \otimes \) be polynomial multiplication.

- pmul: \( (a, b) \to (r, s) \) where \( r s = a \otimes b \), i.e. \( r \) is the upper word of the result, and \( s \) is the lower word.
- pdiv.mod: \( (a, b) \to (q, r) \) where \( q \otimes b = a \oplus r \), i.e. \( q \) is the quotient and \( r \) is the remainder.

It's been a while since I last studied abstract algebra, so I might not have gotten all my terminology correct, but you get the idea.

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