Function guff_ida::cauchy_matrix

pub fn cauchy_matrix<G>(
    field: &G, 
    key: &Vec<G::E>, 
    k: usize, 
    n: usize
) -> Vec<G::E> where
    G: GaloisField, 

Cauchy-form matrix generated from a key

                k columns

    |     1        1             1     |
    |  -------  -------  ...  -------  |
    |  x1 + y1  x1 + y2       x1 + yk  |
    |                                  |
    |     1        1             1     |
    |  -------  -------  ...  -------  |
    |  x2 + y1  x2 + y2       x2 + yk  |  n rows
    |                                  |
    |     :        :      :      :     |
    |                                  |
    |     1        1             1     |
    |  -------  -------  ...  -------  |
    |  xn + y1  xn + y2       xn + yk  |

All [y1 .. yk, x1 .. xn] field values must be distinct non-zero values

TODO: Check that all input values are distinct. Can put all elements on a heap and then check that no parent node has a child node equal to it. Alternatively, check the condition as we’re building the heap? That should work, too.

TODO: use a random number number generator to select k + n distinct field elements (eg, Floyd’s algorithm for shuffling/selection from an array of all field elements if the field size is small, or a modification of the heap approach above for when it’s impractical to list all the field elements)

I’ve ordered the elements with y values first since these will be reused across all rows. I will allow passing in a vector that has more x values than are required

We don’t operate on k, n to produce field values, so they can be passed in as regular types