gh-37595: Simplify computation of all points for curves over finite fields
This PR removes the call to `_points_fast_sqrt()` for computing the set
of all points as it (seems) to always be slower than using
`_points_via_group_structure()`.
While performing this change, I also attempted to clean up the code and
simplify / clarify certain docstrings.
## Justification
When a curve is defined over a prime order finite field with order
larger than 50, the set of all points is computed using the group
structure of the curve.
However, when a curve is implemented over a non-prime field or a field
with order less than 50, the set of all points is instead computed by
brute-force enumeration by checking all x-coordinates.
For all cases I have tested, this enumeration is slower.
### Small characteristic, prime field
When `p < 50`:
```py
sage: E = EllipticCurve(GF(11), [1,1])
sage: %timeit E._points_via_group_structure()
165 µs ± 8.19 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops
each)
sage: %timeit E._points_fast_sqrt()
366 µs ± 34.2 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
sage: sorted(E._points_via_group_structure()) ==
sorted(E._points_fast_sqrt())
True
```
When `p > 50`
```py
sage: E = EllipticCurve(GF(163), [1,1])
sage: %timeit E._points_via_group_structure()
2.49 ms ± 40.6 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: %timeit E._points_fast_sqrt()
4.1 ms ± 112 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: sorted(E._points_via_group_structure()) ==
sorted(E._points_fast_sqrt())
True
```
### Small characteristic, non-prime field
When the order is smaller than 50:
```py
sage: E = EllipticCurve(GF(2^5), [1,0,1,0,1])
sage: %timeit E._points_via_group_structure()
277 µs ± 10.2 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
sage: %timeit E._points_fast_sqrt()
2.96 ms ± 135 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: sorted(E._points_via_group_structure()) ==
sorted(E._points_fast_sqrt())
True
```
When the order is larger than 50:
```py
sage: E = EllipticCurve(GF(11^3), [1,1])
sage: %timeit E._points_via_group_structure()
15.6 ms ± 1.17 ms per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: %timeit E._points_fast_sqrt()
71.4 ms ± 2.94 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
sage: sorted(E._points_via_group_structure()) ==
sorted(E._points_fast_sqrt())
True
```
## Further comments
- The commit which introduced `_points_fast_sqrt()` as a method
implemented it for hyperelliptic curves over finite fields and then
added `HyperellipticCurve_finite_field` as a parent to
`EllipticCurve_finite_field` to inherit this function. With the removal
of the slower `_points_fast_sqrt()`, this parent inheritance has been
removed.
- `ruff` noticed that `from sage.sets.set import Set` was unused, so I
removed it
### :memo: Checklist
<!-- Put an `x` in all the boxes that apply. -->
- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation accordingly.
URL: https://github.com/sagemath/sage/pull/37595
Reported by: Giacomo Pope
Reviewer(s):
