11.6. Cross-references🔗

HexPolyFp builds on the generic dense polynomials and supplies the prime-field specialization the finite-field libraries use:

• HexPoly is the generic dense-polynomial library. Hex.FpPoly is an abbrev over Hex.DensePoly, so every constructor, arithmetic, evaluation, and Euclidean operation documented in that chapter is inherited at the specialized type. The concrete DivModLaws/GcdLaws the generic Euclidean laws are stated under are discharged here for ZMod64 p.

• HexModArith supplies the ZMod64 p coefficient arithmetic: the machine-word modular add, multiply, and inverse that every operation in this chapter ultimately calls, along with the ZMod64.Bounds/ZMod64.PrimeModulus instances the prime-field operations require.

Downstream, the finite-field libraries consume HexPolyFp directly: HexGFqRing builds the quotient ring `Fₚ[x] / (g)` and HexGFqField promotes it to a field using the inverse laws documented above, each conditioned on irreducibility of the modulus, with the Hex.FpPoly.Irreducible witness produced by a checkable Rabin certificate from HexBerlekamp.