EllipticCurve.random_element for char=2

[malb]

This should work:

sage: k.<a> = GF(2^5)
sage: E = EllipticCurve(k,[k.random_element() for _ in range(5)])
sage: E
Elliptic Curve defined by y^2 + (a^3+1)*x*y + (a^4+a^3+a)*y = x^3 +
(a^4+a^3+a^2+a)*x^2 + (a^4+a^2+a+1)*x + a^2 over Finite Field in a of
size 2^5
sage: E.random_element()
Exception (click to the left for traceback):
...
ZeroDivisionError: division by zero in finite field.

Component: number theory

Issue created by migration from https://trac.sagemath.org/ticket/1119

[malb]

comment:1

The attached patch fixes this (probably in a too naive way).

[sagetrac-mabshoff]

comment:2

Attachment: ell_gf2_random.patch.gz

[robertwb]

comment:3

Attachment: ell_gf2_random2.patch.gz

Given E defined by f(x,y) = 0, the patch assumed that there were always exactly zero or two values of y for every x, which is not true. I've attached a patch fixing this issue.

Also, in the characteristic > 2 case, it never considered the 'negative' square-root. I changed this too.

Otherwise, the patch looks good.

[williamstein]

comment:4

#1119 looks good. It is slow but I don't know if that can be helped.

[sagetrac-mabshoff]

comment:5

Merged in 2.9.rc0.