Perf: Pairing on BN254 using direct Fp12 extension and non-native Eval()

[yelhousni]

Description

Similar to #1312 but for BN254.
This PR refactor methods in fields_bn254 and sw_bn254 using a direct Fp12 extension and non-native Eval() introduced in #1299.

TODO:

• Granger-Scott's cyclotomic squarings.
• Karabina's cyclotomic squarings.
• Torus-based arithmetic over direct Fp6 extension.

Type of change

• New feature (non-breaking change which adds functionality)

How has this been tested?

• New tests corresponding to new methods.
• PairingCheck and FinalExpCheck tests against gnark-crypto.

How has this been benchmarked?

Some circuits proved in a in a BN254-PLONK:

circuit	old (scs)	new (scs)	diff (scs)
e(a,b)	3,534,755	2,030,447	1,504,308
e(a,b) * e(c,d) == 1	3,879,428	2,239,682	1,639,746
ECPAIR-2 precompile*	4,407,402	2,425,399	1,982,003
ECPAIR-4 precompile*	7,815,376	4,181,633	3,633,743

*ECPAIR precompiles include G2 subgroup membership.

Checklist:

• I have performed a self-review of my code
• I have commented my code, particularly in hard-to-understand areas
• I have made corresponding changes to the documentation
• I have added tests that prove my fix is effective or that my feature works
• I did not modify files generated from templates
• golangci-lint does not output errors locally
• New and existing unit tests pass locally with my changes
• Any dependent changes have been merged and published in downstream modules

[ivokub]

Suggested edit:

diff --git a/std/algebra/emulated/sw_bn254/pairing.go b/std/algebra/emulated/sw_bn254/pairing.go
index fdfedafe..d2e95540 100644
--- a/std/algebra/emulated/sw_bn254/pairing.go
+++ b/std/algebra/emulated/sw_bn254/pairing.go
@@ -24,7 +24,6 @@ type Pairing struct {
 	curve  *sw_emulated.Curve[BaseField, ScalarField]
 	g2     *G2
 	bTwist *fields_bn254.E2
-	g2gen  *G2Affine
 }
 
 func NewGTEl(a bn254.GT) GTEl {
@@ -82,15 +81,6 @@ func NewPairing(api frontend.API) (*Pairing, error) {
 	}, nil
 }
 
-func (pr Pairing) generators() *G2Affine {
-	if pr.g2gen == nil {
-		_, _, _, g2gen := bn254.Generators()
-		cg2gen := NewG2AffineFixed(g2gen)
-		pr.g2gen = &cg2gen
-	}
-	return pr.g2gen
-}
-
 // Pair calculates the reduced pairing for a set of points
 // ∏ᵢ e(Pᵢ, Qᵢ).
 //

[ivokub]

See the suggested edits for removing deadcode.

Looks good to me, I didn't go through the formulas, but as the tests are functioning, then I think everything is valid. See only comments about removed examples and tests.

One thing I would do is to create isomorphism maps between the tower and direct extension as explicit methods and use them, as imo it makes it a bit clearer to see what extension is used where.

And I think it would be better to wait for #1340 to be merged to ensure that there are no failing tests for edge cases (which I understood you bypassed?)

[yelhousni]

See the suggested edits for removing deadcode.

Looks good to me, I didn't go through the formulas, but as the tests are functioning, then I think everything is valid. See only comments about removed examples and tests.

One thing I would do is to create isomorphism maps between the tower and direct extension as explicit methods and use them, as imo it makes it a bit clearer to see what extension is used where.

And I think it would be better to wait for #1340 to be merged to ensure that there are no failing tests for edge cases (which I understood you bypassed?)

That makes sense. I pushed a few comments that address your review points. We can wait for #1340.
I'm still thinking whether to implement Karabina and Torus-based direct-Fp6 as it would be only useful for computing the exact value of the pairing, but we only use PairingCheck in our protocols. However torus stuff might come in handy in the future for Ethereum SSLE (ethresearch post).

[ivokub]

That makes sense. I pushed a few comments that address your review points. We can wait for #1340. I'm still thinking whether to implement Karabina and Torus-based direct-Fp6 as it would be only useful for computing the exact value of the pairing, but we only use PairingCheck in our protocols. However torus stuff might come in handy in the future for Ethereum SSLE (ethresearch post).

Yup, I agree that with different extensions computing pairing makes less sense and we'd mostly be interested in pairing check (or multi-pairing check). I guess we can keep it currently as is, but have documentation update that it may not be efficient.

[yelhousni]

@ivokub PR is ready now!