Hello all,

I'm messing around trying to create a toy bls12-381 implementation. In order to create the required tower of fields, I'm doing this:

```
reset()
F = GF(0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab)
g1curve = EllipticCurve(F, [0,4])
K2.<x> = PolynomialRing(F)
F2.<u> = F.extension(x^2+1)
K6.<y> = PolynomialRing(F2)
F6.<v> = F2.extension(y^3 - (u+1))
K12.<z> = PolynomialRing(F6)
F12.<w> = F6.extension(z^2 - v)
```

Such towering is described in multiple places, e.g, Optimal Ate Pairings at the 128-bit Security level (hxxps://hal.archives-ouvertes.fr/hal-01620848/document), Implementing Pairings at the 192-bit Security Level (hxxps://eprint.iacr.org/2012/232.pdf) and Faster Subgroup Checks for BLS12-381 (hxxps://pdfs.semanticscholar.org/f413/bf4f22f682043616261e463abd0fd9fdcc54.pdf).

I am implementing example code given in Guide to Pairing Based Cryptography (hxxps://www.crcpress.com/Guide-to-Pairing-Based-Cryptography/Mrabet-Joye/p/book/9781498729505) that relies on the w $F_p^{12}$ element defined in the above extension tower. However, the last step, of this code appears to take an inordinate amount of time (yet to see it complete).

Taking the cue from faster subgroup checks for bls12, I tried to redefine this as:

```
F12.<w> = F6.extension(x^2 - v)
```

but this fails with a type conversion error.

I've tried to search this site to find out how I might generate F12 directly from F2 as I've seen comments indicating that performance of towers of field extensions is... not great and this is also my experience. I have tried to define Fp12 entirely in terms of x from the first PolynomialRing, but I haven't found a way to try to extend directly from Fp2 to Fp12 yet (I only need Fp12 and its sextic twist. I have seen how to create a homomorphism to embed one elements in another field, but I haven't yet tried to use this to simplify. Can this be done? Is there a way to make this performant?

Apologies for the broken links, I'm not allowed to include links yet.