api.md

API Flint

solve_h90
    INPUT: ζ a n-th root of unity, A a Kummer algebra
    OUTPUT: x a solution in A of H90 for ζ, with x^n = the right constant

norm_r
    INPUT: x an element of the Kummer algebra, (a, b) two integers
    OUTPUT: the right norm N_{b/a} of x

first_coordinate
    INPUT: x an element of the Kummer algebra, ζ a n-th root of unity
    OUTPUT: the first coordinate of x in the base 1⊗ζ

matrices (fq_embed_matrices does this job)
    INPUT: x, y elements of two finite fields
    OUTPUT: M, N two matrices corresponding to the map x|->y and its section

Julia

In Julia there will be global variables

• EMBEDDINGS containing already computed embeddings
• H90_ELEMENTS containing already computed solutions of H90
• ROOTS_UNITY containing the chosen roots of unity

A function embed that will take care of everything:

embed
    INPUT: k, K two finite fields
    OUTPUT: an embedding from k to K
1. construct the Kummer algebras A and B associated to k and K using the roots in
   ROOTS_UNITY
2. Check if there is an embedding from k to K in EMBEDDINGS, if so, return
   it
3. Check if there is a solution alpha of H90 in A, if not compute it using
   solve_h90 and save it in H90_ELEMENTS
4. Check if there is a solution beta of H90 in B, if not compute it using
   solve_h90 and save it in H90_ELEMENTS
5. Compute f(beta), where f is a function using an exponantiation, or a norm, depending
   on the degrees of k and K
6. Compute the first coordinates u and v of alpha and f(beta)
7. Compute the embedding u|->v thanks to the function matrices, save it
   in EMBEDDINGS, and return it