Computing isomorphisms and embeddings of finite fields

fq_zech_embed_gens gen_sub gen_sup minpoly sub_ctx sup_ctx

Given two contexts sub_ctx and sup_ctx, such that degree(sub_ctx) divides degree(sup_ctx), compute:

• an element gen_sub in sub_ctx such that gen_sub generates the finite field defined by sub_ctx,
• its minimal polynomial minpoly,
• a root gen_sup of minpoly inside the field defined by sup_ctx.

These data uniquely define an embedding of sub_ctx into sup_ctx.

_fq_zech_embed_gens_naive gen_sub gen_sup minpoly sub_ctx sup_ctx

Given two contexts sub_ctx and sup_ctx, such that degree(sub_ctx) divides degree(sup_ctx), compute an embedding of sub_ctx into sup_ctx defined as follows:

• gen_sub is the canonical generator of sup_ctx (i.e., the class of \(X\)),
• minpoly is the defining polynomial of sub_ctx,
• gen_sup is a root of minpoly inside the field defined by sup_ctx.

fq_zech_embed_matrices embed project gen_sub sub_ctx gen_sup sup_ctx gen_minpoly

Given:

• two contexts sub_ctx and sup_ctx, of respective degrees \(m\) and \(n\), such that \(m\) divides \(n\);
• a generator gen_sub of sub_ctx, its minimal polynomial gen_minpoly, and a root gen_sup of gen_minpoly in sup_ctx, as returned by fq_zech_embed_gens;

Compute:

• the \(n\times m\) matrix embed mapping gen_sub to gen_sup, and all their powers accordingly;
• an \(m\times n\) matrix project such that project \(\times\) embed is the \(m\times m\) identity matrix.

fq_zech_embed_trace_matrix res basis sub_ctx sup_ctx

Given:

• two contexts sub_ctx and sup_ctx, of degrees \(m\) and \(n\), such that \(m\) divides \(n\);
• an \(n\times m\) matrix basis that maps sub_ctx to an isomorphic subfield in sup_ctx;

Compute the \(m\times n\) matrix of the trace from sup_ctx to sub_ctx.

This matrix is computed as

embed_dual_to_mono_matrix(_, sub_ctx) \(\times\) basist \(\times\) embed_mono_to_dual_matrix(_, sup_ctx)}.

Note: if \(m=n\), basis represents a Frobenius, and the result is its inverse matrix.

fq_zech_embed_composition_matrix matrix gen ctx

Compute the composition matrix of gen.

For an element \(a\in\mathbf{F}_{p^n}\), its composition matrix is the matrix whose columns are \(a^0, a^1, \ldots, a^{n-1}\).

fq_zech_embed_composition_matrix_sub matrix gen ctx trunc

Compute the composition matrix of gen, truncated to trunc columns.

fq_zech_embed_mul_matrix matrix gen ctx

Compute the multiplication matrix of gen.

For an element \(a\) in \(\mathbf{F}_{p^n}=\mathbf{F}_p[x]\), its multiplication matrix is the matrix whose columns are (a, ax, dots, ax^{n-1}).

fq_zech_embed_mono_to_dual_matrix res ctx

Compute the change of basis matrix from the monomial basis of ctx to its dual basis.

fq_zech_embed_dual_to_mono_matrix res ctx

Compute the change of basis matrix from the dual basis of ctx to its monomial basis.

fq_zech_modulus_pow_series_inv res ctx trunc

Compute the power series inverse of the reverse of the modulus of ctx up to \(O(x^\texttt{trunc})\).

fq_zech_modulus_derivative_inv m_prime m_prime_inv ctx

Compute the derivative m_prime of the modulus of ctx as an element of ctx, and its inverse m_prime_inv.