dedekind::algebra::𝔽64 Struct Reference

The Galois field of order 64, \(\mathrm{GF}(2^6) = \mathbb{F}_2[x] / (x^6 + x + 1)\). More...

Public Member Functions
constexpr	𝔽64 (std::uint8_t v) noexcept
constexpr 𝔽64	inverse () const

Public Attributes
std::uint8_t	value

Friends
constexpr friend 𝔽64	operator+ (𝔽64 a, 𝔽64 b) noexcept
constexpr friend 𝔽64	operator- (𝔽64 a, 𝔽64 b) noexcept
constexpr friend 𝔽64	operator- (𝔽64 a) noexcept
constexpr friend 𝔽64	operator* (𝔽64 a, 𝔽64 b) noexcept
constexpr friend 𝔽64	operator/ (𝔽64 a, 𝔽64 b)
constexpr friend bool	operator== (𝔽64 a, 𝔽64 b) noexcept

Detailed Description

The Galois field of order 64, \(\mathrm{GF}(2^6) = \mathbb{F}_2[x] / (x^6 + x + 1)\).

The smallest non-prime finite field: 64 elements of characteristic 2, cyclic multiplicative group \(\mathbb{F}_{64}^{\times}\) of order

1. Each element is a polynomial of degree \(< 6\) in \(\mathbb{F}_2[x]\), packed bit-wise into a single std::uint8_t (bit i holds the coefficient of \(x^i\); the top two bits are always zero).

• Addition is XOR (characteristic two; subtraction coincides, so \(-x = x\)).
• Multiplication is polynomial multiplication modulo the irreducible polynomial \(f(x) = x^6 + x + 1\) (bit pattern 0b1000011 = 0x43). This \(f\) is both irreducible and primitive over \(\mathbb{F}_2\).
• The multiplicative inverse is computed by Fermat's little theorem: \(a^{-1} = a^{62}\) on \(\mathbb{F}_{64}^{\times}\), via square-and-multiply.

The struct wrapper exists to canonicalise the value range to \({0, \ldots, 63}\) (C++ concepts cannot restrict types to a subset of uint8_t values); addition and multiplication both preserve this invariant.

As a carrier, 𝔽64 witnesses IsGaloisField (here) and IsVectorSpace<𝔽64, bool> (in algebra:vectorspace). The vector-space claim uses the natural embedding \(\mathbb{F}_2 \hookrightarrow \mathbb{F}_{64}\) as the scalar action on coordinates.

Reference polynomial: Lidl & Niederreiter, Finite Fields, 2nd ed. (Cambridge University Press, 1997), Table C.1.

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The documentation for this struct was generated from the following file:

• src/main/modules/dedekind/algebra/galois.cppm