Fix(emulated pairing): edge cases in torus-based final exp #613

yelhousni · 2023-03-31T16:38:13Z

fixes #605

Lookup2 is slighly more inefficient than select, but because we do select for all E12 elements, then should add up.

ivokub

Made some modifications to avoid passing API around. This made the interfaces simpler as we do not need to initialize the field emulation everywhere.

Also store api in pairing context so that can use select instead of lookup2 in final exp.

I'm not sure if the small comment is correct. Left a remark just in case. Otherwise I think is good to go!

std/algebra/emulated/sw_bn254/pairing.go

std/algebra/emulated/sw_bls12381/pairing.go

yelhousni · 2023-04-01T00:36:10Z

@ivokub last commit does a switch on the Miller loop size (1 vs. product). In the case of 1, the edge-cases do not happen as the pairing is non-degenerate (≠1) so we save the Select/IsZero logic (~13k in BN254).

ivokub · 2023-04-03T08:28:40Z

Suggested edit:

diff --git a/std/algebra/emulated/sw_bls12381/pairing.go b/std/algebra/emulated/sw_bls12381/pairing.go
index 44b345e9..788f6203 100644
--- a/std/algebra/emulated/sw_bls12381/pairing.go
+++ b/std/algebra/emulated/sw_bls12381/pairing.go
@@ -64,25 +64,65 @@ func NewPairing(api frontend.API) (*Pairing, error) {
 	}, nil
 }
 
-// FinalExponentiation computes the exponentiation (∏ᵢ zᵢ)ᵈ
-// where d = (p¹²-1)/r = (p¹²-1)/Φ₁₂(p) ⋅ Φ₁₂(p)/r = (p⁶-1)(p²+1)(p⁴ - p² +1)/r
-// we use instead d=s ⋅ (p⁶-1)(p²+1)(p⁴ - p² +1)/r
-// where s is the cofactor 3 (Hayashida et al.)
-func (pr Pairing) FinalExponentiation(e *GTEl, n int) *GTEl {
+// FinalExponentiation computes the exponentiation (∏ᵢ zᵢ)ᵈ where
+//
+//	d = (p¹²-1)/r = (p¹²-1)/Φ₁₂(p) ⋅ Φ₁₂(p)/r = (p⁶-1)(p²+1)(p⁴ - p² +1)/r
+//
+// we use instead
+//
+//	d=s ⋅ (p⁶-1)(p²+1)(p⁴ - p² +1)/r
+//
+// where s is the cofactor 3 (Hayashida et al.).
+//
+// This is the safe version of the method where e may be {-1,1}. If it is known
+// that e ≠ {-1,1} then using the unsafe version of the method saves
+// considerable amount of constraints. When called with the result of
+// [MillerLoop], then current method is applicable when length of the inputs to
+// Miller loop is 1.
+func (pr Pairing) FinalExponentiation(e *GTEl) *GTEl {
+	return pr.finalExponentiation(e, false)
+}
+
+// FinalExponentiationUnsafe computes the exponentiation (∏ᵢ zᵢ)ᵈ where
+//
+//	d = (p¹²-1)/r = (p¹²-1)/Φ₁₂(p) ⋅ Φ₁₂(p)/r = (p⁶-1)(p²+1)(p⁴ - p² +1)/r
+//
+// we use instead
+//
+//	d=s ⋅ (p⁶-1)(p²+1)(p⁴ - p² +1)/r
+//
+// where s is the cofactor 3 (Hayashida et al.).
+//
+// This is the unsafe version of the method where e may NOT be {-1,1}. If e ∈
+// {-1, 1}, then there exists no valid solution to the circuit. This method is
+// applicable when called with the result of [MillerLoop] method when the length
+// of the inputs to Miller loop is 1.
+func (pr Pairing) FinalExponentiationUnsafe(e *GTEl) *GTEl {
+	return pr.finalExponentiation(e, true)
+}
+
+// finalExponentiation computes the exponentiation (∏ᵢ zᵢ)ᵈ where
+//
+//	d = (p¹²-1)/r = (p¹²-1)/Φ₁₂(p) ⋅ Φ₁₂(p)/r = (p⁶-1)(p²+1)(p⁴ - p² +1)/r
+//
+// we use instead
+//
+//	d=s ⋅ (p⁶-1)(p²+1)(p⁴ - p² +1)/r
+//
+// where s is the cofactor 3 (Hayashida et al.).
+func (pr Pairing) finalExponentiation(e *GTEl, unsafe bool) *GTEl {
 
 	// 1. Easy part
 	// (p⁶-1)(p²+1)
 	var selector1, selector2 frontend.Variable
 	_dummy := pr.Ext6.One()
 
-	switch n {
-	case 1:
+	if unsafe {
 		// The Miller loop result is ≠ {-1,1}, otherwise this means P and Q are
 		// linearly dependant and not from G1 and G2 respectively.
 		// So e ∈ G_{q,2} \ {-1,1} and hence e.C1 ≠ 0.
 		// Nothing to do.
-
-	default:
+	} else {
 		// However, for a product of Miller loops (n>=2) this might happen.  If this is
 		// the case, the result is 1 in the torus. We assign a dummy value (1) to e.C1
 		// and proceed further.
@@ -125,13 +165,12 @@ func (pr Pairing) FinalExponentiation(e *GTEl, n int) *GTEl {
 	t1 = pr.MulTorus(t1, t0)
 
 	var result GTEl
-	switch n {
 	// MulTorus(c, t1) requires c ≠ -t1. When c = -t1, it means the
 	// product is 1 in the torus.
-	case 1:
+	if unsafe {
 		// For a single pairing, this does not happen because the pairing is non-degenerate.
 		result = *pr.DecompressTorus(pr.MulTorus(c, t1))
-	default:
+	} else {
 		// For a product of pairings this might happen when the result is expected to be 1.
 		// We assign a dummy value (1) to t1 and proceed furhter.
 		// Finally we do a select on both edge cases:
@@ -160,11 +199,11 @@ type lineEvaluation struct {
 //
 // This function doesn't check that the inputs are in the correct subgroups.
 func (pr Pairing) Pair(P []*G1Affine, Q []*G2Affine) (*GTEl, error) {
-	res, n, err := pr.MillerLoop(P, Q)
+	res, err := pr.MillerLoop(P, Q)
 	if err != nil {
 		return nil, fmt.Errorf("miller loop: %w", err)
 	}
-	res = pr.FinalExponentiation(res, n)
+	res = pr.finalExponentiation(res, len(P) == 1)
 	return res, nil
 }
 
@@ -202,11 +241,11 @@ var loopCounter = [64]int8{
 
 // MillerLoop computes the multi-Miller loop
 // ∏ᵢ { fᵢ_{u,Q}(P) }
-func (pr Pairing) MillerLoop(P []*G1Affine, Q []*G2Affine) (*GTEl, int, error) {
+func (pr Pairing) MillerLoop(P []*G1Affine, Q []*G2Affine) (*GTEl, error) {
 	// check input size match
 	n := len(P)
 	if n == 0 || n != len(Q) {
-		return nil, n, errors.New("invalid inputs sizes")
+		return nil, errors.New("invalid inputs sizes")
 	}
 
 	res := pr.Ext12.One()
@@ -322,7 +361,7 @@ func (pr Pairing) MillerLoop(P []*G1Affine, Q []*G2Affine) (*GTEl, int, error) {
 	// negative x₀
 	res = pr.Ext12.Conjugate(res)
 
-	return res, n, nil
+	return res, nil
 }
 
 // doubleAndAddStep doubles p1 and adds p2 to the result in affine coordinates, and evaluates the line in Miller loop
diff --git a/std/algebra/emulated/sw_bls12381/pairing_test.go b/std/algebra/emulated/sw_bls12381/pairing_test.go
index 5523ced8..f9b16663 100644
--- a/std/algebra/emulated/sw_bls12381/pairing_test.go
+++ b/std/algebra/emulated/sw_bls12381/pairing_test.go
@@ -35,7 +35,7 @@ func (c *FinalExponentiationCircuit) Define(api frontend.API) error {
 	if err != nil {
 		return fmt.Errorf("new pairing: %w", err)
 	}
-	res := pairing.FinalExponentiation(&c.InGt, 1)
+	res := pairing.FinalExponentiation(&c.InGt)
 	pairing.AssertIsEqual(res, &c.Res)
 	return nil
 }
diff --git a/std/algebra/emulated/sw_bn254/pairing.go b/std/algebra/emulated/sw_bn254/pairing.go
index 4bfaa389..88245f17 100644
--- a/std/algebra/emulated/sw_bn254/pairing.go
+++ b/std/algebra/emulated/sw_bn254/pairing.go
@@ -74,22 +74,62 @@ func NewPairing(api frontend.API) (*Pairing, error) {
 //
 // and r does NOT divide d'
 //
-// FinalExponentiation returns a decompressed element in E12
-func (pr Pairing) FinalExponentiation(e *GTEl, n int) *GTEl {
+// FinalExponentiation returns a decompressed element in E12.
+//
+// This is the safe version of the method where e may be {-1,1}. If it is known
+// that e ≠ {-1,1} then using the unsafe version of the method saves
+// considerable amount of constraints. When called with the result of
+// [MillerLoop], then current method is applicable when length of the inputs to
+// Miller loop is 1.
+func (pr Pairing) FinalExponentiation(e *GTEl) *GTEl {
+	return pr.finalExponentiation(e, false)
+}
+
+// FinalExponentiationUnsafe computes the exponentiation eᵈ where
+//
+//	d = (p¹²-1)/r = (p¹²-1)/Φ₁₂(p) ⋅ Φ₁₂(p)/r = (p⁶-1)(p²+1)(p⁴ - p² +1)/r.
+//
+// We use instead d'= s ⋅ d, where s is the cofactor
+//
+//	2x₀(6x₀²+3x₀+1)
+//
+// and r does NOT divide d'
+//
+// FinalExponentiationUnsafe returns a decompressed element in E12.
+//
+// This is the unsafe version of the method where e may NOT be {-1,1}. If e ∈
+// {-1, 1}, then there exists no valid solution to the circuit. This method is
+// applicable when called with the result of [MillerLoop] method when the length
+// of the inputs to Miller loop is 1.
+func (pr Pairing) FinalExponentiationUnsafe(e *GTEl) *GTEl {
+	return pr.finalExponentiation(e, true)
+}
+
+// finalExponentiation computes the exponentiation eᵈ where
+//
+//	d = (p¹²-1)/r = (p¹²-1)/Φ₁₂(p) ⋅ Φ₁₂(p)/r = (p⁶-1)(p²+1)(p⁴ - p² +1)/r.
+//
+// We use instead d'= s ⋅ d, where s is the cofactor
+//
+//	2x₀(6x₀²+3x₀+1)
+//
+// and r does NOT divide d'
+//
+// finalExponentiation returns a decompressed element in E12
+func (pr Pairing) finalExponentiation(e *GTEl, unsafe bool) *GTEl {
 
 	// 1. Easy part
 	// (p⁶-1)(p²+1)
 	var selector1, selector2 frontend.Variable
 	_dummy := pr.Ext6.One()
 
-	switch n {
-	case 1:
+	if unsafe {
 		// The Miller loop result is ≠ {-1,1}, otherwise this means P and Q are
 		// linearly dependant and not from G1 and G2 respectively.
 		// So e ∈ G_{q,2} \ {-1,1} and hence e.C1 ≠ 0.
 		// Nothing to do.
 
-	default:
+	} else {
 		// However, for a product of Miller loops (n>=2) this might happen.  If this is
 		// the case, the result is 1 in the torus. We assign a dummy value (1) to e.C1
 		// and proceed further.
@@ -138,13 +178,12 @@ func (pr Pairing) FinalExponentiation(e *GTEl, n int) *GTEl {
 	t2 = pr.FrobeniusCubeTorus(t2)
 
 	var result GTEl
-	switch n {
 	// MulTorus(t0, t2) requires t0 ≠ -t2. When t0 = -t2, it means the
 	// product is 1 in the torus.
-	case 1:
+	if unsafe {
 		// For a single pairing, this does not happen because the pairing is non-degenerate.
 		result = *pr.DecompressTorus(pr.MulTorus(t2, t0))
-	default:
+	} else {
 		// For a product of pairings this might happen when the result is expected to be 1.
 		// We assign a dummy value (1) to t0 and proceed furhter.
 		// Finally we do a select on both edge cases:
@@ -165,11 +204,11 @@ func (pr Pairing) FinalExponentiation(e *GTEl, n int) *GTEl {
 //
 // This function doesn't check that the inputs are in the correct subgroups.
 func (pr Pairing) Pair(P []*G1Affine, Q []*G2Affine) (*GTEl, error) {
-	res, n, err := pr.MillerLoop(P, Q)
+	res, err := pr.MillerLoop(P, Q)
 	if err != nil {
 		return nil, fmt.Errorf("miller loop: %w", err)
 	}
-	res = pr.FinalExponentiation(res, n)
+	res = pr.finalExponentiation(res, len(P) == 1)
 	return res, nil
 }
 
@@ -215,11 +254,11 @@ type lineEvaluation struct {
 
 // MillerLoop computes the multi-Miller loop
 // ∏ᵢ { fᵢ_{6x₀+2,Q}(P) · ℓᵢ_{[6x₀+2]Q,π(Q)}(P) · ℓᵢ_{[6x₀+2]Q+π(Q),-π²(Q)}(P) }
-func (pr Pairing) MillerLoop(P []*G1Affine, Q []*G2Affine) (*GTEl, int, error) {
+func (pr Pairing) MillerLoop(P []*G1Affine, Q []*G2Affine) (*GTEl, error) {
 	// check input size match
 	n := len(P)
 	if n == 0 || n != len(Q) {
-		return nil, n, errors.New("invalid inputs sizes")
+		return nil, errors.New("invalid inputs sizes")
 	}
 
 	res := pr.Ext12.One()
@@ -389,7 +428,7 @@ func (pr Pairing) MillerLoop(P []*G1Affine, Q []*G2Affine) (*GTEl, int, error) {
 			}
 
 		default:
-			return nil, n, errors.New("invalid loopCounter")
+			return nil, errors.New("invalid loopCounter")
 		}
 	}
 
@@ -428,7 +467,7 @@ func (pr Pairing) MillerLoop(P []*G1Affine, Q []*G2Affine) (*GTEl, int, error) {
 
 	}
 
-	return res, n, nil
+	return res, nil
 }
 
 // doubleAndAddStep doubles p1 and adds p2 to the result in affine coordinates, and evaluates the line in Miller loop
diff --git a/std/algebra/emulated/sw_bn254/pairing_test.go b/std/algebra/emulated/sw_bn254/pairing_test.go
index 9a00b1db..3d24335e 100644
--- a/std/algebra/emulated/sw_bn254/pairing_test.go
+++ b/std/algebra/emulated/sw_bn254/pairing_test.go
@@ -35,7 +35,7 @@ func (c *FinalExponentiationCircuit) Define(api frontend.API) error {
 	if err != nil {
 		return fmt.Errorf("new pairing: %w", err)
 	}
-	res := pairing.FinalExponentiation(&c.InGt, 1)
+	res := pairing.FinalExponentiation(&c.InGt)
 	pairing.AssertIsEqual(res, &c.Res)
 	return nil
 }

ivokub · 2023-04-03T08:30:19Z

See what do you think about using safe<>unsafe versions of the method instead of passing n into FinalExponentiation method as an argument.

If seems ok, then I can push as commits.

yelhousni · 2023-04-04T07:13:08Z

See what do you think about using safe<>unsafe versions of the method instead of passing n into FinalExponentiation method as an argument.

If seems ok, then I can push as commits.

Yup that seems ok. You can push the commit for both bn254 and bls12-381. Maybe we can also add the safe vs. unsafe test:

diff --git a/std/algebra/emulated/sw_bn254/pairing_test.go b/std/algebra/emulated/sw_bn254/pairing_test.go
index 9a00b1db..3ecfb588 100644
--- a/std/algebra/emulated/sw_bn254/pairing_test.go
+++ b/std/algebra/emulated/sw_bn254/pairing_test.go
@@ -35,8 +35,10 @@ func (c *FinalExponentiationCircuit) Define(api frontend.API) error {
        if err != nil {
                return fmt.Errorf("new pairing: %w", err)
        }
-       res := pairing.FinalExponentiation(&c.InGt, 1)
-       pairing.AssertIsEqual(res, &c.Res)
+       res1 := pairing.FinalExponentiation(&c.InGt)
+       res2 := pairing.FinalExponentiationUnsafe(&c.InGt)
+       pairing.AssertIsEqual(res1, &c.Res)
+       pairing.AssertIsEqual(res2, &c.Res)
        return nil
 }

ivokub · 2023-04-04T09:17:37Z

Added safe<>unsafe versions.
added both safe and unsafe test for final exp test
added separate test case which should fail with unsafe but not with safe.

yelhousni · 2023-04-04T14:17:52Z

awesome! thanks for that. It's good on my side, I think we can merge this.

ivokub · 2023-04-04T14:18:34Z

awesome! thanks for that. It's good on my side, I think we can merge this.

All good from my side. You are good to merge!

yelhousni added 6 commits March 31, 2023 13:02

feat(fields_bn254): add IsZero in extensions

465cec1

feat(fields_bn254): add Select in extensions

7af74dd

feat(fields_bn254): add String helpers

dbd819c

fix: finalExp when element is 1 in torus

773a4b9

fix(bn254/pairing): issue 605, edge cases in torus arithmetic

71e742b

fix(bls12-381/pairing): issue 605, edge cases in torus arithmetic

10e6b9d

yelhousni requested a review from ivokub March 31, 2023 16:38

yelhousni self-assigned this Mar 31, 2023

yelhousni added the zk-evm label Mar 31, 2023

yelhousni added this to the v0.9.0 milestone Mar 31, 2023

yelhousni mentioned this pull request Mar 31, 2023

BN254 Pairing issue #605

Closed

yelhousni and others added 9 commits March 31, 2023 18:52

test(bls12-381/pairing): fix test

39a87ec

chore: remove debug printing code

efcaa09

refactor: embed api and init emulation in tower

04885e0

refactor: do not pass api in towers

74e12de

refactor: do not pass api in pairing

710c17c

fix: one omitted change

998938c

feat: store api in pairing structs

12c89e6

refactor: reuse dummy one

b756271

refactor: use select instead of lookup2

c9bba78

Lookup2 is slighly more inefficient than select, but because we do select for all E12 elements, then should add up.

ivokub approved these changes Mar 31, 2023

View reviewed changes

std/algebra/emulated/sw_bn254/pairing.go Outdated Show resolved Hide resolved

std/algebra/emulated/sw_bls12381/pairing.go Outdated Show resolved Hide resolved

yelhousni added 3 commits April 1, 2023 01:30

docs: correct comment

27dce2b

docs: clarify some comments

5683295

perf(pairings): switch to no edge-cases when single pairing

d4f67b6

feat: add PairingCheck function

de7b2ec

refactor: separate final exp into safe and unsafe

dd6918f

ivokub added 2 commits April 4, 2023 11:05

test: test also unsafe final exp

7f37089

test: add safe final exp tests

5b819fe

yelhousni merged commit 6939cf5 into develop Apr 4, 2023

yelhousni deleted the fix-605 branch April 4, 2023 14:20

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Fix(emulated pairing): edge cases in torus-based final exp #613

Fix(emulated pairing): edge cases in torus-based final exp #613

yelhousni commented Mar 31, 2023

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ivokub commented Apr 4, 2023

Fix(emulated pairing): edge cases in torus-based final exp #613

Fix(emulated pairing): edge cases in torus-based final exp #613

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