CECCO::Isomorphism< A, B > Class Template Reference

Functor representing the field isomorphism φ: A → B between two same-size finite fields. More...

#include <fields.hpp>

Public Member Functions
	Isomorphism ()
	Construct (or retrieve from cache) the isomorphism map A → B.
constexpr	Isomorphism (const std::vector< size_t > &iso)
	Construct from a precomputed mapping table (iso[i] = φ(A(i))).
constexpr B	operator() (const A &a) const
	Apply φ to a.
constexpr Isomorphism< B, A >	inverse () const
	Inverse isomorphism φ⁻¹: B → A.

Detailed Description

template<FiniteFieldType A, FiniteFieldType B>
requires Isomorphic<A, B>
class CECCO::Isomorphism< A, B >

Functor representing the field isomorphism φ: A → B between two same-size finite fields.

Template Parameters

A	Source finite field
B	Target finite field (Isomorphic<A, B>)

The isomorphism is built deterministically: a Conway polynomial of the prime field gives generators α ∈ A and β ∈ B as common roots, and φ is extended linearly via a change-of-basis matrix over the prime subfield. The resulting table is stored in a single details::IsomorphismPair shared between Isomorphism<A, B> and Isomorphism<B, A>. It is a field homomorphism: φ(a + b) = φ(a) + φ(b), φ(a · b) = φ(a) · φ(b), φ(0) = 0, φ(1) = 1.

Usage_Example

using F2    = Fp<2>;
using F4    = Ext<F2, {1, 1, 1}>;
using F16_a = Ext<F4, {2, 1, 1}>;
using F16_b = Ext<F4, {1, 2, 1}>;
 
Isomorphism<F16_a, F16_b> phi;
F16_a a(4);
F16_b b = phi(a);
F16_a c = phi.inverse()(b);
assert(a == c);

Definition at line 1562 of file fields.hpp.

Constructor & Destructor Documentation

Isomorphism() [1/2]

template<FiniteFieldType A, FiniteFieldType B>

CECCO::Isomorphism< A, B >::Isomorphism

(

)

Construct (or retrieve from cache) the isomorphism map A → B.

Definition at line 1597 of file fields.hpp.

Isomorphism() [2/2]

template<FiniteFieldType A, FiniteFieldType B>

CECCO::Isomorphism< A, B >::Isomorphism ( const std::vector< size_t > & iso )

inlineconstexpr

Construct from a precomputed mapping table (iso[i] = φ(A(i))).

Warning

No validation — incorrect input gives undefined behaviour. Internal use.

Definition at line 1574 of file fields.hpp.

Member Function Documentation

inverse()

template<FiniteFieldType A, FiniteFieldType B>

Isomorphism< B, A > CECCO::Isomorphism< A, B >::inverse ( ) const

constexpr

Inverse isomorphism φ⁻¹: B → A.

Definition at line 1621 of file fields.hpp.

operator()()

template<FiniteFieldType A, FiniteFieldType B>

B CECCO::Isomorphism< A, B >::operator() ( const A & a ) const

inlineconstexpr

Apply φ to a.

Definition at line 1577 of file fields.hpp.

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

• fields.hpp