Lighter construction of finite field elements from lists

[remyoudompheng]

📚 Description

When doing intensive polynomial arithmetic with the NTL implementation the constructor with lists is called a large number of times and may spend a lot of time constructing the vector_space and FreeModuleElement objects.

The very common call to vector_space(map=False) is optimized to be as cheap as possible using the already cached object.

The common case of lists of length 0 and 1 is replaced by cheaper shortcuts.

This improves performance when doing intensive polynomial computations over finite field extensions.

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

[remyoudompheng]

Elliptic curve isogenies provide examples for a few scenarios:

• polynomial with many zeros, most coefficients belong to the prime subfield
• polynomial with many zeros
• polynomial with few zeros

Benchmark before patch:

sage: p = next_prime(2**128)
sage: K = GF(p**2, 'a')
sage: E = EllipticCurve(K, [0, 1])
sage: pol = E.torsion_polynomial(97).monic()
sage: %timeit E.isogeny(pol, check=False)
1.3 s ± 4.56 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

sage: E = EllipticCurve(K, [0, K.gen()])
sage: pol = E.torsion_polynomial(97).monic()
sage: %timeit E.isogeny(pol, check=False)
2.38 s ± 6 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

sage: E = EllipticCurve(K, [K.gen(), K.gen()])
sage: pol = E.torsion_polynomial(97).monic()
sage: %timeit E.isogeny(pol, check=False)
4.78 s ± 8.36 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

After patch:

sage: p = next_prime(2**128)
sage: K = GF(p**2, 'a')
sage: E = EllipticCurve(K, [0, 1])
sage: pol = E.torsion_polynomial(97).monic()
sage: %timeit E.isogeny(pol, check=False)
602 ms ± 1.58 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

sage: E = EllipticCurve(K, [0, K.gen()])
sage: pol = E.torsion_polynomial(97).monic()
sage: %timeit E.isogeny(pol, check=False)
779 ms ± 4.41 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

sage: E = EllipticCurve(K, [K.gen(), K.gen()])
sage: pol = E.torsion_polynomial(97).monic()
sage: %timeit E.isogeny(pol, check=False)
1.48 s ± 13.1 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

[tscrim]

LGTM (assuming the tests pass).