diff --git a/ecc/bls12-377/g1_test.go b/ecc/bls12-377/g1_test.go
index f32180292..442b68e74 100644
--- a/ecc/bls12-377/g1_test.go
+++ b/ecc/bls12-377/g1_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG1AffineEndomorphism(t *testing.T) {
+func TestG1Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -60,7 +60,7 @@ func TestG1AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineIsOnCurve(t *testing.T) {
+func TestIsOnG1(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -109,7 +109,7 @@ func TestG1AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineConversions(t *testing.T) {
+func TestG1Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -499,7 +499,7 @@ func TestG1AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineCofactorCleaning(t *testing.T) {
+func TestG1CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -538,7 +538,7 @@ func TestG1AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG1AffineBatchScalarMultiplication(t *testing.T) {
+func TestG1BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/bls12-377/g2_test.go b/ecc/bls12-377/g2_test.go
index eba5190a5..32ca456e7 100644
--- a/ecc/bls12-377/g2_test.go
+++ b/ecc/bls12-377/g2_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG2AffineEndomorphism(t *testing.T) {
+func TestG2Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -74,7 +74,7 @@ func TestG2AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineIsOnCurve(t *testing.T) {
+func TestIsOnG2(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -123,7 +123,7 @@ func TestG2AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineConversions(t *testing.T) {
+func TestG2Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -491,7 +491,7 @@ func TestG2AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineCofactorCleaning(t *testing.T) {
+func TestG2CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -527,7 +527,7 @@ func TestG2AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG2AffineBatchScalarMultiplication(t *testing.T) {
+func TestG2BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/bls12-381/g1.go b/ecc/bls12-381/g1.go
index 6f3e639d1..5f5a94311 100644
--- a/ecc/bls12-381/g1.go
+++ b/ecc/bls12-381/g1.go
@@ -506,6 +506,19 @@ func (p *G1Jac) IsInSubGroup() bool {
 
 }
 
+func GeneratePointNotInG1(f fp.Element) G1Jac {
+	var res, jac G1Jac
+	aff := MapToCurve1(&f)
+	g1Isogeny(&aff)
+	jac.FromAffine(&aff)
+	// p+x²ϕ(p) = [r]p
+	res.phi(&jac).
+		mulBySeed(&res).
+		mulBySeed(&res).
+		AddAssign(&jac)
+	return res
+}
+
 // mulWindowed computes the 2-bits windowed double-and-add scalar
 // multiplication p=[s]q in Jacobian coordinates.
 func (p *G1Jac) mulWindowed(q *G1Jac, s *big.Int) *G1Jac {
diff --git a/ecc/bls12-381/g1_test.go b/ecc/bls12-381/g1_test.go
index c705b7a67..5f50bd339 100644
--- a/ecc/bls12-381/g1_test.go
+++ b/ecc/bls12-381/g1_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG1AffineEndomorphism(t *testing.T) {
+func TestG1Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -60,7 +60,7 @@ func TestG1AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineIsOnCurve(t *testing.T) {
+func TestIsOnG1(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -105,11 +105,18 @@ func TestG1AffineIsOnCurve(t *testing.T) {
 		},
 		GenFp(),
 	))
+	properties.Property("[BLS12-381] IsInSubGroup should return false for a point on the cofactor-torsion", prop.ForAll(
+		func(a fp.Element) bool {
+			op := fuzzCofactorOfG1(a)
+			return op.IsOnCurve() && !op.IsInSubGroup()
+		},
+		GenFp(),
+	))
 
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineConversions(t *testing.T) {
+func TestG1Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -499,7 +506,7 @@ func TestG1AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineCofactorCleaning(t *testing.T) {
+func TestG1CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -538,7 +545,7 @@ func TestG1AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG1AffineBatchScalarMultiplication(t *testing.T) {
+func TestG1BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -852,6 +859,18 @@ func BenchmarkG1AffineDouble(b *testing.B) {
 		a.Double(&a)
 	}
 }
+func fuzzCofactorOfG1(f fp.Element) G1Jac {
+	var res, jac G1Jac
+	aff := MapToCurve1(&f)
+	g1Isogeny(&aff)
+	jac.FromAffine(&aff)
+	// p+x²ϕ(p) = [r]p
+	res.phi(&jac).
+		mulBySeed(&res).
+		mulBySeed(&res).
+		AddAssign(&jac)
+	return res
+}
 
 func fuzzG1Jac(p *G1Jac, f fp.Element) G1Jac {
 	var res G1Jac
diff --git a/ecc/bls12-381/g2.go b/ecc/bls12-381/g2.go
index b92fcbadb..7c8551d70 100644
--- a/ecc/bls12-381/g2.go
+++ b/ecc/bls12-381/g2.go
@@ -503,6 +503,18 @@ func (p *G2Jac) IsInSubGroup() bool {
 	return res.Equal(&img)
 }
 
+func GeneratePointNotInG2(f E2) G2Jac {
+	var res, jac G2Jac
+	aff := MapToCurve2(&f)
+	g2Isogeny(&aff)
+	jac.FromAffine(&aff)
+	// ψ(p)-[x₀]P = [r]p
+	res.mulBySeed(&jac)
+	jac.psi(&jac)
+	res.AddAssign(&jac)
+	return res
+}
+
 // mulWindowed computes the 2-bits windowed double-and-add scalar
 // multiplication p=[s]q in Jacobian coordinates.
 func (p *G2Jac) mulWindowed(q *G2Jac, s *big.Int) *G2Jac {
diff --git a/ecc/bls12-381/g2_test.go b/ecc/bls12-381/g2_test.go
index 45753354c..4ae25ba8f 100644
--- a/ecc/bls12-381/g2_test.go
+++ b/ecc/bls12-381/g2_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG2AffineEndomorphism(t *testing.T) {
+func TestG2Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -74,7 +74,7 @@ func TestG2AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineIsOnCurve(t *testing.T) {
+func TestIsOnG2(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -119,11 +119,18 @@ func TestG2AffineIsOnCurve(t *testing.T) {
 		},
 		GenE2(),
 	))
+	properties.Property("[BLS12-381] IsInSubGroup should return false for a point on the cofactor-torsion", prop.ForAll(
+		func(a fptower.E2) bool {
+			op := fuzzCofactorOfG2(a)
+			return op.IsOnCurve() && !op.IsInSubGroup()
+		},
+		GenE2(),
+	))
 
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineConversions(t *testing.T) {
+func TestG2Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -491,7 +498,7 @@ func TestG2AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineCofactorCleaning(t *testing.T) {
+func TestG2CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -527,7 +534,7 @@ func TestG2AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG2AffineBatchScalarMultiplication(t *testing.T) {
+func TestG2BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -841,6 +848,17 @@ func BenchmarkG2AffineDouble(b *testing.B) {
 		a.Double(&a)
 	}
 }
+func fuzzCofactorOfG2(f fptower.E2) G2Jac {
+	var res, jac G2Jac
+	aff := MapToCurve2(&f)
+	g2Isogeny(&aff)
+	jac.FromAffine(&aff)
+	// ψ(p)-[x₀]P = [r]p
+	res.mulBySeed(&jac)
+	jac.psi(&jac)
+	res.AddAssign(&jac)
+	return res
+}
 
 func fuzzG2Jac(p *G2Jac, f fptower.E2) G2Jac {
 	var res G2Jac
diff --git a/ecc/bls24-315/g1_test.go b/ecc/bls24-315/g1_test.go
index 8cfd54e14..9a5003907 100644
--- a/ecc/bls24-315/g1_test.go
+++ b/ecc/bls24-315/g1_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG1AffineEndomorphism(t *testing.T) {
+func TestG1Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -60,7 +60,7 @@ func TestG1AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineIsOnCurve(t *testing.T) {
+func TestIsOnG1(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -109,7 +109,7 @@ func TestG1AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineConversions(t *testing.T) {
+func TestG1Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -499,7 +499,7 @@ func TestG1AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineCofactorCleaning(t *testing.T) {
+func TestG1CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -538,7 +538,7 @@ func TestG1AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG1AffineBatchScalarMultiplication(t *testing.T) {
+func TestG1BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/bls24-315/g2_test.go b/ecc/bls24-315/g2_test.go
index b748c00f6..1dec0cd03 100644
--- a/ecc/bls24-315/g2_test.go
+++ b/ecc/bls24-315/g2_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG2AffineEndomorphism(t *testing.T) {
+func TestG2Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -74,7 +74,7 @@ func TestG2AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineIsOnCurve(t *testing.T) {
+func TestIsOnG2(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -123,7 +123,7 @@ func TestG2AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineConversions(t *testing.T) {
+func TestG2Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -491,7 +491,7 @@ func TestG2AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineCofactorCleaning(t *testing.T) {
+func TestG2CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -527,7 +527,7 @@ func TestG2AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG2AffineBatchScalarMultiplication(t *testing.T) {
+func TestG2BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/bls24-317/g1_test.go b/ecc/bls24-317/g1_test.go
index 647f6a6c4..8ff80d660 100644
--- a/ecc/bls24-317/g1_test.go
+++ b/ecc/bls24-317/g1_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG1AffineEndomorphism(t *testing.T) {
+func TestG1Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -60,7 +60,7 @@ func TestG1AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineIsOnCurve(t *testing.T) {
+func TestIsOnG1(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -109,7 +109,7 @@ func TestG1AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineConversions(t *testing.T) {
+func TestG1Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -499,7 +499,7 @@ func TestG1AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineCofactorCleaning(t *testing.T) {
+func TestG1CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -538,7 +538,7 @@ func TestG1AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG1AffineBatchScalarMultiplication(t *testing.T) {
+func TestG1BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/bls24-317/g2_test.go b/ecc/bls24-317/g2_test.go
index 8d4e7aca3..ffb56ab7e 100644
--- a/ecc/bls24-317/g2_test.go
+++ b/ecc/bls24-317/g2_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG2AffineEndomorphism(t *testing.T) {
+func TestG2Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -74,7 +74,7 @@ func TestG2AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineIsOnCurve(t *testing.T) {
+func TestIsOnG2(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -123,7 +123,7 @@ func TestG2AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineConversions(t *testing.T) {
+func TestG2Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -491,7 +491,7 @@ func TestG2AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineCofactorCleaning(t *testing.T) {
+func TestG2CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -527,7 +527,7 @@ func TestG2AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG2AffineBatchScalarMultiplication(t *testing.T) {
+func TestG2BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/bn254/g1_test.go b/ecc/bn254/g1_test.go
index a489fec09..d97c98dd6 100644
--- a/ecc/bn254/g1_test.go
+++ b/ecc/bn254/g1_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG1AffineEndomorphism(t *testing.T) {
+func TestG1Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -60,7 +60,7 @@ func TestG1AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineIsOnCurve(t *testing.T) {
+func TestIsOnG1(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -109,7 +109,7 @@ func TestG1AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineConversions(t *testing.T) {
+func TestG1Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -499,7 +499,7 @@ func TestG1AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineBatchScalarMultiplication(t *testing.T) {
+func TestG1BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/bn254/g2.go b/ecc/bn254/g2.go
index 50fd2806c..c8f31ad92 100644
--- a/ecc/bn254/g2.go
+++ b/ecc/bn254/g2.go
@@ -516,6 +516,24 @@ func (p *G2Jac) IsInSubGroup() bool {
 	return res.Equal(&c)
 }
 
+func GeneratePointNotInG2(f E2) G2Jac {
+	var res, jac, a, b, c G2Jac
+	aff := MapToCurve2(&f)
+	jac.FromAffine(&aff)
+	// [x₀+1]P + ψ([x₀]P) + ψ²([x₀]P) - ψ³([2x₀]P) = [r]P
+	a.mulBySeed(&jac)
+	b.psi(&a)
+	a.AddAssign(&jac)
+	res.psi(&b)
+	c.Set(&res).
+		AddAssign(&b).
+		AddAssign(&a)
+	res.psi(&res).
+		Double(&res).
+		SubAssign(&c)
+	return res
+}
+
 // mulWindowed computes the 2-bits windowed double-and-add scalar
 // multiplication p=[s]q in Jacobian coordinates.
 func (p *G2Jac) mulWindowed(q *G2Jac, s *big.Int) *G2Jac {
diff --git a/ecc/bn254/g2_test.go b/ecc/bn254/g2_test.go
index eb6b67164..6737ce4d3 100644
--- a/ecc/bn254/g2_test.go
+++ b/ecc/bn254/g2_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG2AffineEndomorphism(t *testing.T) {
+func TestG2Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -73,7 +73,7 @@ func TestG2AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineIsOnCurve(t *testing.T) {
+func TestIsOnG2(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -118,11 +118,18 @@ func TestG2AffineIsOnCurve(t *testing.T) {
 		},
 		GenE2(),
 	))
+	properties.Property("[BN254] IsInSubGroup should return false for a point on the cofactor-torsion", prop.ForAll(
+		func(a fptower.E2) bool {
+			op := fuzzCofactorOfG2(a)
+			return op.IsOnCurve() && !op.IsInSubGroup()
+		},
+		GenE2(),
+	))
 
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineConversions(t *testing.T) {
+func TestG2Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -490,7 +497,7 @@ func TestG2AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineCofactorCleaning(t *testing.T) {
+func TestG2CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -526,7 +533,7 @@ func TestG2AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG2AffineBatchScalarMultiplication(t *testing.T) {
+func TestG2BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -840,6 +847,23 @@ func BenchmarkG2AffineDouble(b *testing.B) {
 		a.Double(&a)
 	}
 }
+func fuzzCofactorOfG2(f fptower.E2) G2Jac {
+	var res, jac, a, b, c G2Jac
+	aff := MapToCurve2(&f)
+	jac.FromAffine(&aff)
+	// [x₀+1]P + ψ([x₀]P) + ψ²([x₀]P) - ψ³([2x₀]P) = [r]P
+	a.mulBySeed(&jac)
+	b.psi(&a)
+	a.AddAssign(&jac)
+	res.psi(&b)
+	c.Set(&res).
+		AddAssign(&b).
+		AddAssign(&a)
+	res.psi(&res).
+		Double(&res).
+		SubAssign(&c)
+	return res
+}
 
 func fuzzG2Jac(p *G2Jac, f fptower.E2) G2Jac {
 	var res G2Jac
diff --git a/ecc/bw6-633/g1_test.go b/ecc/bw6-633/g1_test.go
index 59d10ffea..4a33d0362 100644
--- a/ecc/bw6-633/g1_test.go
+++ b/ecc/bw6-633/g1_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG1AffineEndomorphism(t *testing.T) {
+func TestG1Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -60,7 +60,7 @@ func TestG1AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineIsOnCurve(t *testing.T) {
+func TestIsOnG1(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -109,7 +109,7 @@ func TestG1AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineConversions(t *testing.T) {
+func TestG1Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -499,7 +499,7 @@ func TestG1AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineCofactorCleaning(t *testing.T) {
+func TestG1CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -538,7 +538,7 @@ func TestG1AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG1AffineBatchScalarMultiplication(t *testing.T) {
+func TestG1BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/bw6-633/g2_test.go b/ecc/bw6-633/g2_test.go
index 0d3ca6317..f957217a6 100644
--- a/ecc/bw6-633/g2_test.go
+++ b/ecc/bw6-633/g2_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG2AffineEndomorphism(t *testing.T) {
+func TestG2Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -60,7 +60,7 @@ func TestG2AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineIsOnCurve(t *testing.T) {
+func TestIsOnG2(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -109,7 +109,7 @@ func TestG2AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineConversions(t *testing.T) {
+func TestG2Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -469,7 +469,7 @@ func TestG2AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineCofactorCleaning(t *testing.T) {
+func TestG2CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -508,7 +508,7 @@ func TestG2AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG2AffineBatchScalarMultiplication(t *testing.T) {
+func TestG2BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/bw6-761/g1_test.go b/ecc/bw6-761/g1_test.go
index fd81b0ffe..55374561f 100644
--- a/ecc/bw6-761/g1_test.go
+++ b/ecc/bw6-761/g1_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG1AffineEndomorphism(t *testing.T) {
+func TestG1Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -60,7 +60,7 @@ func TestG1AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineIsOnCurve(t *testing.T) {
+func TestIsOnG1(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -109,7 +109,7 @@ func TestG1AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineConversions(t *testing.T) {
+func TestG1Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -499,7 +499,7 @@ func TestG1AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineCofactorCleaning(t *testing.T) {
+func TestG1CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -538,7 +538,7 @@ func TestG1AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG1AffineBatchScalarMultiplication(t *testing.T) {
+func TestG1BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/bw6-761/g2_test.go b/ecc/bw6-761/g2_test.go
index 92555681b..d4fb02f4d 100644
--- a/ecc/bw6-761/g2_test.go
+++ b/ecc/bw6-761/g2_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG2AffineEndomorphism(t *testing.T) {
+func TestG2Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -60,7 +60,7 @@ func TestG2AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineIsOnCurve(t *testing.T) {
+func TestIsOnG2(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -109,7 +109,7 @@ func TestG2AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineConversions(t *testing.T) {
+func TestG2Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -469,7 +469,7 @@ func TestG2AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG2AffineCofactorCleaning(t *testing.T) {
+func TestG2CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -508,7 +508,7 @@ func TestG2AffineCofactorCleaning(t *testing.T) {
 
 }
 
-func TestG2AffineBatchScalarMultiplication(t *testing.T) {
+func TestG2BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/grumpkin/g1_test.go b/ecc/grumpkin/g1_test.go
index bb1111e97..0f311a21e 100644
--- a/ecc/grumpkin/g1_test.go
+++ b/ecc/grumpkin/g1_test.go
@@ -18,7 +18,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG1AffineEndomorphism(t *testing.T) {
+func TestG1Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -60,7 +60,7 @@ func TestG1AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineIsOnCurve(t *testing.T) {
+func TestIsOnG1(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -109,7 +109,7 @@ func TestG1AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineConversions(t *testing.T) {
+func TestG1Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -499,7 +499,7 @@ func TestG1AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineBatchScalarMultiplication(t *testing.T) {
+func TestG1BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/ecc/secp256k1/g1_test.go b/ecc/secp256k1/g1_test.go
index aee07c948..2800422c2 100644
--- a/ecc/secp256k1/g1_test.go
+++ b/ecc/secp256k1/g1_test.go
@@ -20,7 +20,7 @@ import (
 	"github.com/leanovate/gopter/prop"
 )
 
-func TestG1AffineEndomorphism(t *testing.T) {
+func TestG1Endomorphism(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -62,7 +62,7 @@ func TestG1AffineEndomorphism(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineIsOnCurve(t *testing.T) {
+func TestIsOnG1(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -111,7 +111,7 @@ func TestG1AffineIsOnCurve(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineConversions(t *testing.T) {
+func TestG1Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -501,7 +501,7 @@ func TestG1AffineOps(t *testing.T) {
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
-func TestG1AffineBatchScalarMultiplication(t *testing.T) {
+func TestG1BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
diff --git a/internal/generator/ecc/template/point.go.tmpl b/internal/generator/ecc/template/point.go.tmpl
index fd7a4f3db..b10f3ac67 100644
--- a/internal/generator/ecc/template/point.go.tmpl
+++ b/internal/generator/ecc/template/point.go.tmpl
@@ -810,6 +810,51 @@ func (p *{{ $TJacobian }}) IsOnCurve() bool {
 {{- end}}
 
 
+{{- if eq .Name "bls12-381"}}
+    {{- if eq .PointName "g1" }}
+func GeneratePointNotIn{{ toUpper .PointName}}(f {{ .CoordType}}) {{ $TJacobian }} {
+	var res, jac {{ $TJacobian }}
+	aff := MapToCurve1(&f)
+	g1Isogeny(&aff)
+	jac.FromAffine(&aff)
+    // p+x²ϕ(p) = [r]p
+	res.phi(&jac).
+		mulBySeed(&res).
+		mulBySeed(&res).
+		AddAssign(&jac)
+    {{- else}}
+func GeneratePointNotIn{{ toUpper .PointName}}(f E2) {{ $TJacobian }} {
+	var res, jac {{ $TJacobian }}
+	aff := MapToCurve2(&f)
+	g2Isogeny(&aff)
+	jac.FromAffine(&aff)
+    // ψ(p)-[x₀]P = [r]p
+	res.mulBySeed(&jac)
+	jac.psi(&jac)
+    res.AddAssign(&jac)
+    {{- end}}
+	return res
+}
+{{- else if and (eq .Name "bn254") (eq .PointName "g2")}}
+func GeneratePointNotIn{{ toUpper .PointName}}(f E2) {{ $TJacobian }} {
+	var res, jac, a, b, c {{ $TJacobian }}
+   	aff := MapToCurve2(&f)
+	jac.FromAffine(&aff)
+    // [x₀+1]P + ψ([x₀]P) + ψ²([x₀]P) - ψ³([2x₀]P) = [r]P
+	a.mulBySeed(&jac)
+	b.psi(&a)
+	a.AddAssign(&jac)
+	res.psi(&b)
+	c.Set(&res).
+		AddAssign(&b).
+		AddAssign(&a)
+	res.psi(&res).
+		Double(&res).
+        SubAssign(&c)
+	return res
+}
+{{- end}}
+
 // mulWindowed computes the 2-bits windowed double-and-add scalar
 // multiplication p=[s]q in Jacobian coordinates.
 func (p *{{ $TJacobian }}) mulWindowed(q *{{ $TJacobian }}, s *big.Int) *{{ $TJacobian }} {
diff --git a/internal/generator/ecc/template/tests/point.go.tmpl b/internal/generator/ecc/template/tests/point.go.tmpl
index 1cbc95303..5d763a066 100644
--- a/internal/generator/ecc/template/tests/point.go.tmpl
+++ b/internal/generator/ecc/template/tests/point.go.tmpl
@@ -36,7 +36,7 @@ import (
 )
 
 {{if .GLV}}
-    func Test{{ $TAffine }}Endomorphism(t *testing.T) {
+    func Test{{ toUpper .PointName }}Endomorphism(t *testing.T) {
 		t.Parallel()
         parameters := gopter.DefaultTestParameters()
         if testing.Short() {
@@ -104,7 +104,7 @@ import (
     }
 {{end}}
 
-func Test{{ $TAffine }}IsOnCurve(t *testing.T) {
+func TestIsOn{{ toUpper .PointName }}(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -150,11 +150,21 @@ func Test{{ $TAffine }}IsOnCurve(t *testing.T) {
 		{{$fuzzer}},
 	))
 
+    {{- if or (eq .Name "bls12-381") (and (eq .Name "bn254") (eq .PointName "g2"))}}
+	properties.Property("[{{ toUpper .Name }}] IsInSubGroup should return false for a point on the cofactor-torsion", prop.ForAll(
+		func(a {{ .CoordType}}) bool {
+			op := fuzzCofactorOf{{ toUpper .PointName}}(a)
+			return op.IsOnCurve() && !op.IsInSubGroup()
+		},
+		{{$fuzzer}},
+	))
+    {{- end}}
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
 
-func Test{{ $TAffine }}Conversions(t *testing.T) {
+func Test{{ toUpper .PointName }}Conversions(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -569,7 +579,7 @@ func Test{{ $TAffine }}Ops(t *testing.T) {
 
 
 {{if .CofactorCleaning }}
-func Test{{ $TAffine }}CofactorCleaning(t *testing.T) {
+func Test{{ toUpper .PointName }}CofactorClearing(t *testing.T) {
 	t.Parallel()
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -621,7 +631,7 @@ func Test{{ $TAffine }}CofactorCleaning(t *testing.T) {
 }
 {{end}}
 
-func Test{{ $TAffine }}BatchScalarMultiplication(t *testing.T) {
+func Test{{ toUpper .PointName }}BatchScalarMultiplication(t *testing.T) {
 
 	parameters := gopter.DefaultTestParameters()
 	if testing.Short() {
@@ -941,6 +951,50 @@ func Benchmark{{ toUpper .PointName}}AffineDouble(b *testing.B) {
 	}
 }
 
+{{- if eq .Name "bls12-381"}}
+func fuzzCofactorOf{{ toUpper .PointName}}(f {{ .CoordType}}) {{ $TJacobian }} {
+	var res, jac {{ $TJacobian }}
+    {{- if eq .PointName "g1" }}
+	aff := MapToCurve1(&f)
+	g1Isogeny(&aff)
+	jac.FromAffine(&aff)
+    // p+x²ϕ(p) = [r]p
+	res.phi(&jac).
+		mulBySeed(&res).
+		mulBySeed(&res).
+		AddAssign(&jac)
+    {{- else}}
+	aff := MapToCurve2(&f)
+	g2Isogeny(&aff)
+	jac.FromAffine(&aff)
+    // ψ(p)-[x₀]P = [r]p
+	res.mulBySeed(&jac)
+	jac.psi(&jac)
+    res.AddAssign(&jac)
+    {{- end}}
+	return res
+}
+
+{{- else if and (eq .Name "bn254") (eq .PointName "g2")}}
+func fuzzCofactorOf{{ toUpper .PointName}}(f {{ .CoordType}}) {{ $TJacobian }} {
+	var res, jac, a, b, c {{ $TJacobian }}
+   	aff := MapToCurve2(&f)
+	jac.FromAffine(&aff)
+    // [x₀+1]P + ψ([x₀]P) + ψ²([x₀]P) - ψ³([2x₀]P) = [r]P
+	a.mulBySeed(&jac)
+	b.psi(&a)
+	a.AddAssign(&jac)
+	res.psi(&b)
+	c.Set(&res).
+		AddAssign(&b).
+		AddAssign(&a)
+	res.psi(&res).
+		Double(&res).
+        SubAssign(&c)
+	return res
+}
+{{- end}}
+
 func fuzz{{ $TJacobian }}(p *{{ $TJacobian }}, f {{ .CoordType}}) {{ $TJacobian }} {
 	var res {{ $TJacobian }}
 	res.X.Mul(&p.X, &f).Mul(&res.X, &f)