Trait FieldExtensionAlgebra

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pub trait FieldExtensionAlgebra<Base: FieldAlgebra>:
    FieldAlgebra
    + From<Base>
    + Add<Base, Output = Self>
    + AddAssign<Base>
    + Sub<Base, Output = Self>
    + SubAssign<Base>
    + Mul<Base, Output = Self>
    + MulAssign<Base> {
    const D: usize;

    // Required methods
    fn from_base(b: Base) -> Self;
    fn from_base_slice(bs: &[Base]) -> Self;
    fn from_base_fn<F: FnMut(usize) -> Base>(f: F) -> Self;
    fn from_base_iter<I: Iterator<Item = Base>>(iter: I) -> Self;
    fn as_base_slice(&self) -> &[Base];

    // Provided method
    fn monomial(exponent: usize) -> Self { ... }
}

Expand description

A commutative algebra over an extension field.

Mathematically, this trait captures a slightly more interesting structure than the above one liner. As implemented here, A FieldExtensionAlgebra FEA over and extension field EF is really the result of applying extension of scalars to a FieldAlgebra FA to lift FA from an algebra over F to an algebra over EF and so FEA = EF ⊗ FA where the tensor product is over F.

Required Associated Constants§

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const D: usize

Required Methods§

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fn from_base(b: Base) -> Self

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fn from_base_slice(bs: &[Base]) -> Self

Suppose this field extension is represented by the quotient ring B[X]/(f(X)) where B is Base and f is an irreducible polynomial of degree D. This function takes a slice bs of length at exactly D, and constructs the field element \sum_i bs[i] * X^i.

NB: The value produced by this function fundamentally depends on the choice of irreducible polynomial f. Care must be taken to ensure portability if these values might ever be passed to (or rederived within) another compilation environment where a different f might have been used.

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fn from_base_fn<F: FnMut(usize) -> Base>(f: F) -> Self

Similar to core:array::from_fn, with the same caveats as from_base_slice.

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fn from_base_iter<I: Iterator<Item = Base>>(iter: I) -> Self

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fn as_base_slice(&self) -> &[Base]

Suppose this field extension is represented by the quotient ring B[X]/(f(X)) where B is Base and f is an irreducible polynomial of degree D. This function takes a field element \sum_i bs[i] * X^i and returns the coefficients as a slice bs of length at most D containing, from lowest degree to highest.

NB: The value produced by this function fundamentally depends on the choice of irreducible polynomial f. Care must be taken to ensure portability if these values might ever be passed to (or rederived within) another compilation environment where a different f might have been used.

Provided Methods§

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fn monomial(exponent: usize) -> Self

Suppose this field extension is represented by the quotient ring B[X]/(f(X)) where B is Base and f is an irreducible polynomial of degree D. This function returns the field element X^exponent if exponent < D and panics otherwise. (The fact that f is not known at the point that this function is defined prevents implementing exponentiation of higher powers since the reduction cannot be performed.)

NB: The value produced by this function fundamentally depends on the choice of irreducible polynomial f. Care must be taken to ensure portability if these values might ever be passed to (or rederived within) another compilation environment where a different f might have been used.