diff --git a/ecc/bls12-377/fp/element_test.go b/ecc/bls12-377/fp/element_test.go
index 2a1971116..430d42106 100644
--- a/ecc/bls12-377/fp/element_test.go
+++ b/ecc/bls12-377/fp/element_test.go
@@ -457,6 +457,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bls12-377/fr/element_test.go b/ecc/bls12-377/fr/element_test.go
index a8dc8c14a..ba4b6856a 100644
--- a/ecc/bls12-377/fr/element_test.go
+++ b/ecc/bls12-377/fr/element_test.go
@@ -453,6 +453,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bls12-377/fr/poseidon2/hash.go b/ecc/bls12-377/fr/poseidon2/hash.go
index 631cd02da..57bf3621b 100644
--- a/ecc/bls12-377/fr/poseidon2/hash.go
+++ b/ecc/bls12-377/fr/poseidon2/hash.go
@@ -17,7 +17,7 @@ import (
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{GetDefaultParameters()}, make([]byte, fr.Bytes))
+		NewDefaultPermutation(), make([]byte, fr.Bytes))
 }
 
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
diff --git a/ecc/bls12-377/fr/poseidon2/poseidon2.go b/ecc/bls12-377/fr/poseidon2/poseidon2.go
index 2a631437d..4c1208b71 100644
--- a/ecc/bls12-377/fr/poseidon2/poseidon2.go
+++ b/ecc/bls12-377/fr/poseidon2/poseidon2.go
@@ -142,6 +142,12 @@ func NewPermutationWithSeed(t, rf, rp int, seed string) *Permutation {
 	return res
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this curve.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // sBox applies the sBox on buffer[index]
 func (h *Permutation) sBox(index int, input []fr.Element) {
 	var tmp fr.Element
diff --git a/ecc/bls12-381/fp/element_test.go b/ecc/bls12-381/fp/element_test.go
index d1339ef8d..c5e9cd5cd 100644
--- a/ecc/bls12-381/fp/element_test.go
+++ b/ecc/bls12-381/fp/element_test.go
@@ -457,6 +457,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bls12-381/fr/element_test.go b/ecc/bls12-381/fr/element_test.go
index 0172785b7..61a0e2602 100644
--- a/ecc/bls12-381/fr/element_test.go
+++ b/ecc/bls12-381/fr/element_test.go
@@ -453,6 +453,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bls12-381/fr/poseidon2/hash.go b/ecc/bls12-381/fr/poseidon2/hash.go
index c5788f80d..ab568bcec 100644
--- a/ecc/bls12-381/fr/poseidon2/hash.go
+++ b/ecc/bls12-381/fr/poseidon2/hash.go
@@ -17,7 +17,7 @@ import (
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{GetDefaultParameters()}, make([]byte, fr.Bytes))
+		NewDefaultPermutation(), make([]byte, fr.Bytes))
 }
 
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
diff --git a/ecc/bls12-381/fr/poseidon2/poseidon2.go b/ecc/bls12-381/fr/poseidon2/poseidon2.go
index fddff1f45..31fb07507 100644
--- a/ecc/bls12-381/fr/poseidon2/poseidon2.go
+++ b/ecc/bls12-381/fr/poseidon2/poseidon2.go
@@ -142,6 +142,12 @@ func NewPermutationWithSeed(t, rf, rp int, seed string) *Permutation {
 	return res
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this curve.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // sBox applies the sBox on buffer[index]
 func (h *Permutation) sBox(index int, input []fr.Element) {
 	var tmp fr.Element
diff --git a/ecc/bls24-315/fp/element_test.go b/ecc/bls24-315/fp/element_test.go
index a4186669f..f3ffdeadf 100644
--- a/ecc/bls24-315/fp/element_test.go
+++ b/ecc/bls24-315/fp/element_test.go
@@ -455,6 +455,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bls24-315/fr/element_test.go b/ecc/bls24-315/fr/element_test.go
index 25eb19d2a..2faec8ce6 100644
--- a/ecc/bls24-315/fr/element_test.go
+++ b/ecc/bls24-315/fr/element_test.go
@@ -453,6 +453,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bls24-315/fr/poseidon2/hash.go b/ecc/bls24-315/fr/poseidon2/hash.go
index 23285b149..d8334b99c 100644
--- a/ecc/bls24-315/fr/poseidon2/hash.go
+++ b/ecc/bls24-315/fr/poseidon2/hash.go
@@ -17,7 +17,7 @@ import (
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{GetDefaultParameters()}, make([]byte, fr.Bytes))
+		NewDefaultPermutation(), make([]byte, fr.Bytes))
 }
 
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
diff --git a/ecc/bls24-315/fr/poseidon2/poseidon2.go b/ecc/bls24-315/fr/poseidon2/poseidon2.go
index 2076b7f07..ccc0b650e 100644
--- a/ecc/bls24-315/fr/poseidon2/poseidon2.go
+++ b/ecc/bls24-315/fr/poseidon2/poseidon2.go
@@ -142,6 +142,12 @@ func NewPermutationWithSeed(t, rf, rp int, seed string) *Permutation {
 	return res
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this curve.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // sBox applies the sBox on buffer[index]
 func (h *Permutation) sBox(index int, input []fr.Element) {
 	var tmp fr.Element
diff --git a/ecc/bls24-317/fp/element_test.go b/ecc/bls24-317/fp/element_test.go
index e9de2cefa..b6cd95f89 100644
--- a/ecc/bls24-317/fp/element_test.go
+++ b/ecc/bls24-317/fp/element_test.go
@@ -455,6 +455,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bls24-317/fr/element_test.go b/ecc/bls24-317/fr/element_test.go
index 974e54b86..61e492dba 100644
--- a/ecc/bls24-317/fr/element_test.go
+++ b/ecc/bls24-317/fr/element_test.go
@@ -453,6 +453,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bls24-317/fr/poseidon2/hash.go b/ecc/bls24-317/fr/poseidon2/hash.go
index e416f0615..6d5908818 100644
--- a/ecc/bls24-317/fr/poseidon2/hash.go
+++ b/ecc/bls24-317/fr/poseidon2/hash.go
@@ -17,7 +17,7 @@ import (
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{GetDefaultParameters()}, make([]byte, fr.Bytes))
+		NewDefaultPermutation(), make([]byte, fr.Bytes))
 }
 
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
diff --git a/ecc/bls24-317/fr/poseidon2/poseidon2.go b/ecc/bls24-317/fr/poseidon2/poseidon2.go
index 6801ab065..42b44b02a 100644
--- a/ecc/bls24-317/fr/poseidon2/poseidon2.go
+++ b/ecc/bls24-317/fr/poseidon2/poseidon2.go
@@ -142,6 +142,12 @@ func NewPermutationWithSeed(t, rf, rp int, seed string) *Permutation {
 	return res
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this curve.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // sBox applies the sBox on buffer[index]
 func (h *Permutation) sBox(index int, input []fr.Element) {
 	var tmp fr.Element
diff --git a/ecc/bn254/fp/element_test.go b/ecc/bn254/fp/element_test.go
index c593258d7..a9a82ade6 100644
--- a/ecc/bn254/fp/element_test.go
+++ b/ecc/bn254/fp/element_test.go
@@ -453,6 +453,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bn254/fr/element_test.go b/ecc/bn254/fr/element_test.go
index eb958912d..5784f7a73 100644
--- a/ecc/bn254/fr/element_test.go
+++ b/ecc/bn254/fr/element_test.go
@@ -453,6 +453,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bn254/fr/poseidon2/hash.go b/ecc/bn254/fr/poseidon2/hash.go
index 2930ce251..9a291ae86 100644
--- a/ecc/bn254/fr/poseidon2/hash.go
+++ b/ecc/bn254/fr/poseidon2/hash.go
@@ -17,7 +17,7 @@ import (
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{GetDefaultParameters()}, make([]byte, fr.Bytes))
+		NewDefaultPermutation(), make([]byte, fr.Bytes))
 }
 
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
diff --git a/ecc/bn254/fr/poseidon2/poseidon2.go b/ecc/bn254/fr/poseidon2/poseidon2.go
index e9831ee66..40a49dbd2 100644
--- a/ecc/bn254/fr/poseidon2/poseidon2.go
+++ b/ecc/bn254/fr/poseidon2/poseidon2.go
@@ -142,6 +142,12 @@ func NewPermutationWithSeed(t, rf, rp int, seed string) *Permutation {
 	return res
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this curve.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // sBox applies the sBox on buffer[index]
 func (h *Permutation) sBox(index int, input []fr.Element) {
 	var tmp fr.Element
diff --git a/ecc/bw6-633/fp/element_test.go b/ecc/bw6-633/fp/element_test.go
index 683bcdd0a..dd5fe6701 100644
--- a/ecc/bw6-633/fp/element_test.go
+++ b/ecc/bw6-633/fp/element_test.go
@@ -465,6 +465,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bw6-633/fr/element_test.go b/ecc/bw6-633/fr/element_test.go
index 21e76b150..010cfbd16 100644
--- a/ecc/bw6-633/fr/element_test.go
+++ b/ecc/bw6-633/fr/element_test.go
@@ -455,6 +455,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bw6-633/fr/poseidon2/hash.go b/ecc/bw6-633/fr/poseidon2/hash.go
index f5eb25338..6a4cfddea 100644
--- a/ecc/bw6-633/fr/poseidon2/hash.go
+++ b/ecc/bw6-633/fr/poseidon2/hash.go
@@ -17,7 +17,7 @@ import (
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{GetDefaultParameters()}, make([]byte, fr.Bytes))
+		NewDefaultPermutation(), make([]byte, fr.Bytes))
 }
 
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
diff --git a/ecc/bw6-633/fr/poseidon2/poseidon2.go b/ecc/bw6-633/fr/poseidon2/poseidon2.go
index b3a90546a..51b26364b 100644
--- a/ecc/bw6-633/fr/poseidon2/poseidon2.go
+++ b/ecc/bw6-633/fr/poseidon2/poseidon2.go
@@ -142,6 +142,12 @@ func NewPermutationWithSeed(t, rf, rp int, seed string) *Permutation {
 	return res
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this curve.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // sBox applies the sBox on buffer[index]
 func (h *Permutation) sBox(index int, input []fr.Element) {
 	var tmp fr.Element
diff --git a/ecc/bw6-761/fp/element_test.go b/ecc/bw6-761/fp/element_test.go
index 4419103b6..40a7d0bd3 100644
--- a/ecc/bw6-761/fp/element_test.go
+++ b/ecc/bw6-761/fp/element_test.go
@@ -469,6 +469,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bw6-761/fr/element_test.go b/ecc/bw6-761/fr/element_test.go
index 36859ef68..3864e60a5 100644
--- a/ecc/bw6-761/fr/element_test.go
+++ b/ecc/bw6-761/fr/element_test.go
@@ -457,6 +457,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/bw6-761/fr/poseidon2/hash.go b/ecc/bw6-761/fr/poseidon2/hash.go
index da5d124ca..93f85eca7 100644
--- a/ecc/bw6-761/fr/poseidon2/hash.go
+++ b/ecc/bw6-761/fr/poseidon2/hash.go
@@ -17,7 +17,7 @@ import (
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{GetDefaultParameters()}, make([]byte, fr.Bytes))
+		NewDefaultPermutation(), make([]byte, fr.Bytes))
 }
 
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
diff --git a/ecc/bw6-761/fr/poseidon2/poseidon2.go b/ecc/bw6-761/fr/poseidon2/poseidon2.go
index 833fd242e..58738b1c3 100644
--- a/ecc/bw6-761/fr/poseidon2/poseidon2.go
+++ b/ecc/bw6-761/fr/poseidon2/poseidon2.go
@@ -142,6 +142,12 @@ func NewPermutationWithSeed(t, rf, rp int, seed string) *Permutation {
 	return res
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this curve.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // sBox applies the sBox on buffer[index]
 func (h *Permutation) sBox(index int, input []fr.Element) {
 	var tmp fr.Element
diff --git a/ecc/grumpkin/fp/element_test.go b/ecc/grumpkin/fp/element_test.go
index 1c858fc8a..50a7d4fec 100644
--- a/ecc/grumpkin/fp/element_test.go
+++ b/ecc/grumpkin/fp/element_test.go
@@ -453,6 +453,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/grumpkin/fr/element_test.go b/ecc/grumpkin/fr/element_test.go
index c99df3572..4cefc59a0 100644
--- a/ecc/grumpkin/fr/element_test.go
+++ b/ecc/grumpkin/fr/element_test.go
@@ -453,6 +453,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/grumpkin/fr/poseidon2/hash.go b/ecc/grumpkin/fr/poseidon2/hash.go
index 5cf8077e7..758b17ccb 100644
--- a/ecc/grumpkin/fr/poseidon2/hash.go
+++ b/ecc/grumpkin/fr/poseidon2/hash.go
@@ -17,7 +17,7 @@ import (
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{GetDefaultParameters()}, make([]byte, fr.Bytes))
+		NewDefaultPermutation(), make([]byte, fr.Bytes))
 }
 
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
diff --git a/ecc/grumpkin/fr/poseidon2/poseidon2.go b/ecc/grumpkin/fr/poseidon2/poseidon2.go
index ff1ac9aa7..3e0d0b9bb 100644
--- a/ecc/grumpkin/fr/poseidon2/poseidon2.go
+++ b/ecc/grumpkin/fr/poseidon2/poseidon2.go
@@ -142,6 +142,12 @@ func NewPermutationWithSeed(t, rf, rp int, seed string) *Permutation {
 	return res
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this curve.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // sBox applies the sBox on buffer[index]
 func (h *Permutation) sBox(index int, input []fr.Element) {
 	var tmp fr.Element
diff --git a/ecc/secp256k1/fp/element_test.go b/ecc/secp256k1/fp/element_test.go
index 074234814..d8b8bd80f 100644
--- a/ecc/secp256k1/fp/element_test.go
+++ b/ecc/secp256k1/fp/element_test.go
@@ -451,6 +451,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/secp256k1/fr/element_test.go b/ecc/secp256k1/fr/element_test.go
index a67a52d9a..1112ffb12 100644
--- a/ecc/secp256k1/fr/element_test.go
+++ b/ecc/secp256k1/fr/element_test.go
@@ -451,6 +451,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/stark-curve/fp/element_test.go b/ecc/stark-curve/fp/element_test.go
index 6f0419b08..d5457b524 100644
--- a/ecc/stark-curve/fp/element_test.go
+++ b/ecc/stark-curve/fp/element_test.go
@@ -453,6 +453,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/ecc/stark-curve/fr/element_test.go b/ecc/stark-curve/fr/element_test.go
index 741adb877..6578dd4ba 100644
--- a/ecc/stark-curve/fr/element_test.go
+++ b/ecc/stark-curve/fr/element_test.go
@@ -453,6 +453,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/field/babybear/element_test.go b/field/babybear/element_test.go
index 06fed55bf..b96929dd8 100644
--- a/field/babybear/element_test.go
+++ b/field/babybear/element_test.go
@@ -398,6 +398,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/field/babybear/poseidon2/hash.go b/field/babybear/poseidon2/hash.go
index cf37f0c12..a7edc2eb2 100644
--- a/field/babybear/poseidon2/hash.go
+++ b/field/babybear/poseidon2/hash.go
@@ -16,13 +16,19 @@ import (
 // NewMerkleDamgardHasher returns a Poseidon2 hasher using the Merkle-Damgard
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
-	params := GetDefaultParameters()
+	p := NewDefaultPermutation()
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{params},
-		make([]byte, params.Width/2*fr.Bytes),
+		p,
+		make([]byte, p.params.Width/2*fr.Bytes),
 	)
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this field.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
 // The default parameters are,
 //
diff --git a/field/generator/internal/templates/element/tests.go b/field/generator/internal/templates/element/tests.go
index 622ff0d5f..ba0ec748c 100644
--- a/field/generator/internal/templates/element/tests.go
+++ b/field/generator/internal/templates/element/tests.go
@@ -475,6 +475,19 @@ func Test{{toTitle .ElementName}}Bytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPair{{.ElementName}}) bool {
+			var b {{.ElementName}}
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/field/generator/internal/templates/poseidon2/hash.go.tmpl b/field/generator/internal/templates/poseidon2/hash.go.tmpl
index 471a4270e..dd7783ec6 100644
--- a/field/generator/internal/templates/poseidon2/hash.go.tmpl
+++ b/field/generator/internal/templates/poseidon2/hash.go.tmpl
@@ -8,13 +8,19 @@ import (
 // NewMerkleDamgardHasher returns a Poseidon2 hasher using the Merkle-Damgard
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
-	params := GetDefaultParameters()
+	p := NewDefaultPermutation()
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{params},
-		make([]byte, params.Width/2*fr.Bytes),
+		p,
+		make([]byte, p.params.Width/2*fr.Bytes),
 	)
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this field.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
 // The default parameters are,
 //
diff --git a/field/goldilocks/element_test.go b/field/goldilocks/element_test.go
index d9bfd73da..f563accdf 100644
--- a/field/goldilocks/element_test.go
+++ b/field/goldilocks/element_test.go
@@ -398,6 +398,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/field/goldilocks/poseidon2/hash.go b/field/goldilocks/poseidon2/hash.go
index 3c4b7456a..d90063940 100644
--- a/field/goldilocks/poseidon2/hash.go
+++ b/field/goldilocks/poseidon2/hash.go
@@ -16,13 +16,19 @@ import (
 // NewMerkleDamgardHasher returns a Poseidon2 hasher using the Merkle-Damgard
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
-	params := GetDefaultParameters()
+	p := NewDefaultPermutation()
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{params},
-		make([]byte, params.Width/2*fr.Bytes),
+		p,
+		make([]byte, p.params.Width/2*fr.Bytes),
 	)
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this field.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
 // The default parameters are,
 //
diff --git a/field/koalabear/element_test.go b/field/koalabear/element_test.go
index 039781aa1..17c0239f1 100644
--- a/field/koalabear/element_test.go
+++ b/field/koalabear/element_test.go
@@ -398,6 +398,19 @@ func TestElementBytes(t *testing.T) {
 		genA,
 	))
 
+	properties.Property("SetBytesCanonical(Bytes()) should stay constant", prop.ForAll(
+		func(a testPairElement) bool {
+			var b Element
+			bytes := a.element.Bytes()
+			if err := b.SetBytesCanonical(bytes[:]); err != nil {
+				t.Error(err)
+				return false
+			}
+			return a.element.Equal(&b)
+		},
+		genA,
+	))
+
 	properties.TestingRun(t, gopter.ConsoleReporter(false))
 }
 
diff --git a/field/koalabear/poseidon2/hash.go b/field/koalabear/poseidon2/hash.go
index e37239348..c05c42017 100644
--- a/field/koalabear/poseidon2/hash.go
+++ b/field/koalabear/poseidon2/hash.go
@@ -16,13 +16,19 @@ import (
 // NewMerkleDamgardHasher returns a Poseidon2 hasher using the Merkle-Damgard
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
-	params := GetDefaultParameters()
+	p := NewDefaultPermutation()
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{params},
-		make([]byte, params.Width/2*fr.Bytes),
+		p,
+		make([]byte, p.params.Width/2*fr.Bytes),
 	)
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this field.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
 // The default parameters are,
 //
diff --git a/field/koalabear/poseidon2/test-vectors.json b/field/koalabear/poseidon2/test-vectors.json
new file mode 100644
index 000000000..4aac2a013
--- /dev/null
+++ b/field/koalabear/poseidon2/test-vectors.json
@@ -0,0 +1,24 @@
+[
+  { "in":  [],
+    "out": [0, 0, 0, 0, 0, 0, 0, 0]
+  },
+  {
+    "in":  [[0, 1, 2, 3, 4, 5, 6, 7]],
+    "out": [1402915949, 2083603674, 612464613, 903296317, 1327519169, 1646568100, 1786812904, 1917132066]
+  },
+  {
+    "in": [
+      [20, 30, 40, 50, 60, 70, 80, 90],
+      [120, 130, 140, 150, 160, 170, 180, 190]
+    ],
+    "out": [1735559108, 1813232198, 1226565528, 1795826654, 232717972, 522260852, 1109013755, 1232040797]
+  },
+  {
+    "in": [
+      [20, 30, 40, 50, 60, 70, 80, 90],
+      [120, 130, 140, 150, 160, 170, 180, 190],
+      [320, 330, 340, 350, 360, 370, 380, 390]
+    ],
+    "out": [2089473715, 1269546008, 32120401, 1497392503, 1194364877, 1059604979, 1458772903, 705747974]
+  }
+]
\ No newline at end of file
diff --git a/field/koalabear/poseidon2/vectors_test.go b/field/koalabear/poseidon2/vectors_test.go
new file mode 100644
index 000000000..0125406b5
--- /dev/null
+++ b/field/koalabear/poseidon2/vectors_test.go
@@ -0,0 +1,58 @@
+package poseidon2
+
+import (
+	"bytes"
+	"encoding/binary"
+	"encoding/json"
+	"os"
+	"testing"
+
+	"github.com/consensys/gnark-crypto/field/koalabear"
+	"github.com/stretchr/testify/require"
+)
+
+const width = 16
+
+type testCase struct {
+	In  [][]uint32 `json:"in"`
+	Out []uint32   `json:"out"`
+}
+
+func TestVectors(t *testing.T) {
+
+	f, err := os.Open("test-vectors.json")
+	require.NoError(t, err)
+	defer f.Close()
+
+	var testCases []testCase
+	require.NoError(t, json.NewDecoder(f).Decode(&testCases))
+
+	hsh := NewMerkleDamgardHasher()
+	require.Equal(t, width, GetDefaultParameters().Width, "unexpected hash width")
+	require.Equal(t, width*koalabear.Bytes, 2*hsh.BlockSize(), "unexpected input block size")
+
+	var buf [width * koalabear.Bytes / 2]byte
+
+	for _, c := range testCases {
+		hsh.Reset()
+		require.Equal(t, width/2, len(c.Out), "unexpected output block size")
+
+		for _, in := range c.In {
+			require.Equal(t, width/2, len(in), "unexpected input block size")
+
+			for i := range in {
+				_, err = binary.Encode(buf[i*koalabear.Bytes:(i+1)*koalabear.Bytes], binary.BigEndian, in[i])
+				require.NoError(t, err)
+			}
+			_, err = hsh.Write(buf[:])
+			require.NoError(t, err)
+		}
+
+		res := bytes.NewReader(hsh.Sum(nil))
+		var out [width / 2]uint32
+		for i := range width / 2 {
+			require.NoError(t, binary.Read(res, binary.BigEndian, &out[i]))
+		}
+		require.Equal(t, c.Out, out[:], "unexpected output on input %v", c.In)
+	}
+}
diff --git a/internal/generator/crypto/hash/poseidon2/template/hash.go.tmpl b/internal/generator/crypto/hash/poseidon2/template/hash.go.tmpl
index 51d7ee945..d25d8229f 100644
--- a/internal/generator/crypto/hash/poseidon2/template/hash.go.tmpl
+++ b/internal/generator/crypto/hash/poseidon2/template/hash.go.tmpl
@@ -9,7 +9,7 @@ import (
 // construction with the default parameters.
 func NewMerkleDamgardHasher() gnarkHash.StateStorer {
 	return gnarkHash.NewMerkleDamgardHasher(
-		&Permutation{GetDefaultParameters()}, make([]byte, fr.Bytes))
+		NewDefaultPermutation(), make([]byte, fr.Bytes))
 }
 
 // GetDefaultParameters returns a set of parameters for the Poseidon2 permutation.
diff --git a/internal/generator/crypto/hash/poseidon2/template/poseidon2.go.tmpl b/internal/generator/crypto/hash/poseidon2/template/poseidon2.go.tmpl
index 94daa9d18..6ebb48dd3 100644
--- a/internal/generator/crypto/hash/poseidon2/template/poseidon2.go.tmpl
+++ b/internal/generator/crypto/hash/poseidon2/template/poseidon2.go.tmpl
@@ -141,6 +141,12 @@ func NewPermutationWithSeed(t, rf, rp int, seed string) *Permutation {
 	return res
 }
 
+// NewDefaultPermutation returns a Poseidon2 permutation with the default
+// recommended parameters for this curve.
+func NewDefaultPermutation() *Permutation {
+	return &Permutation{params: GetDefaultParameters()}
+}
+
 // sBox applies the sBox on buffer[index]
 func (h *Permutation) sBox(index int, input []fr.Element) {
 	var tmp fr.Element